arXiv:2011.01053v1 [math.CO] 2 Nov 2020 .Zri:Dprmn fMteais oaSaeUniversit State Iowa Mathematics, of n Foundation Department Science Israel Zerbib: an S. by supported is Segal-Halevi (1.1) of .SglHlv:Dprmn fCmue cec,AilUni Ariel Science, Computer of Department Israel. Technion, Segal-Halevi: Mathematics, E. of Department Briggs: J. oes(“colors”) covers o vr face every for elkoncniuu eso fteee etrkonSperne 1.1 better-known even Theorem the of version continuous well-known A cover polytope a 1.1. ipe ∆ simplex nescigsub-collection: intersecting etcsare vertices .Aaoi .Bre n .Zri r upre yBFgrant BSF by supported are Zerbib S. and Berger DMS-1953929. E. grant Aharoni, NSF R. by supported is Zerbib .Bre:Dprmn fMteais nvriyo Haifa, of paper. University the Mathematics, of of Department 8 through Berger: 6 E. Sections MegaGran on of worked framework Aharoni the 82374 in no. Government agreement Federation This grant Russian Sklodowska-Curie 2020 Technion. Marie Horizon the Union’s the at European under the Chair from Bank funding Discount received has the and 2023464 no. .Aaoi eateto ahmtc,Tcno,Ire an Israel Technion, Mathematics, of Department [email protected] Aharoni: R. RCINLYBLNE YEGAH N RAINBOW AND HYPERGRAPHS BALANCED FRACTIONALLY ∆ olcino lsdsets closed of collection A ae[1 rvdaribw(.. oofl eso,i hc hr are there which in version, colorful) (i.e., rainbow a proved [11] Gale n utdmninladribwvrin fteKMtheorem. KKM the of versions rainbow and Multidimensional . − 1 O HRN,EIBRE,JSP RGS RLSEGAL-HALEVI, EREL BRIGGS, JOSEPH BERGER, ELI AHARONI, RON If . onso h acignme fsc yegah.Tee in w mat These, rainbow theorem and hypergraphs. Hall’s problems such of division of of mulitple-cake version number on matching topological results constan the has a that on Using set bounds edge its side. on vertex function non-negative a ists Abstract. n d -intervals. − P e 1 i ednt by denote we , i , σ stesto points of set the is TeKMterm[14]) theorem KKM (The of ≤ A A ∆ o hrn sSpotdb h salSineFudto ( Foundation Science Israel the by Supported is Aharoni Ron . v i n v , d n where , priehprrp scalled is hypergraph -partite − 1 then , ∈ V e (∆ K THEOREMS KKM A V i ( T v N HR ZERBIB SHIRA AND 1. j ( n v , P = ) v − ∈ ~ x Introduction 1 t e fvrie.Te( The vertices. of set its ) V σ ) ∈ ( = i , (∆ δ ⊆ ( V n ,j i, x v − [ 1 ∈ (∆ . ∈ 1 1 ,namely ), ) σ x , . . . , Let [ A n A n − v ,adec otiue e othe to set a contributes each and ], v 1 A rcinlybalanced fractionally 6= aifig(.)i alda called is (1.1) satisfying ) v n [email protected] ∅ v , ) . .712/20. o. ∈ Israel. est,Israel. versity, eerhadinvto programme innovation and research .Ti okwsspotdb the by supported was work This 8. e ∈ ,USA. y, i o 7-521-96we Ron when 075-15-2019-1926 no. t R ( IT ogpun,Russia. Dolgoprudny, MIPT, d i V + n 1 = ) o 2016077. no. ae spr fapoetthat project a of part is paper (∆ [email protected] satisfying hnsi families in chings ere neach in degrees t n − [email protected] e , rv lower prove e 1 n ) i ecoe subsets closed be , fteeex- there if ( un yield turn, − [email protected] j ’ em is: lemma r’s for 0 = ) 1)-dimensional P x i S)grant ISF) n .Its 1. = KKM- KKM- . Given j 6= S. . . i . . 2 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
i i Theorem 1.2 (Rainbow KKM [11]). For every i ∈ [n] let A = Av, v ∈ V (∆n−1) be a KKM-cover of ∆n−1. Then there exists a permutation π of V (∆n−1) such that Ai 6= ∅. i∈[n] π(ei)
T (As noted, the vertices of ∆n−1 are ei.) The notion of KKM-covers can be extended to general polytopes. Given a poly- tope Q, a KKM-cover is a collection of closed sets Av defined for every vertex v of Q, satisfying
(1.2) σ ⊆ Av v∈[V (σ) for every face σ of Q. Remark 1.3. In more general versions, a set is assigned to every face, not only to vertices. See Theorem 2.5 below. In this paper we will focus on polytopes Q that are products of simplices. We i i shall prove results of the form “given a family of KKM-covers A = (Av | v ∈ V (Q)), there exists a large rainbow subfamily that has non-empty intersection”. We shall pose a further requirement, that the vertices v participating in the rainbow set are disjoint (in an intuitive sense, to be made precise below) - this is necessary for our main application, multiple cakes division. We shall use two main tools: (1) Theorem 2.4. Given a cake-division problem with d cakes, the theorem produces a (d + 1)-partite hypergraph, with weights on the edges, that are evenly distributed in every vertex side. We shall call such hypergraphs fractionally balanced. The theorem then asserts that the number of agents that can be satisfied by some division of the cakes is at least the matching number of this hypergraph. This theorem is inspired by a result of Meunier and Su [19]. (2) A topological version of Hall’s theorem. It will be used to find lower bounds on the matching numbers of fractionally balanced hypergraphs.
1.2. Division of multiple cakes. The bulk of the paper deals with partitions of multiple cakes. A “cake” in this setting is a copy of the unit interval [0, 1]. A partition of [0, 1] into a interval pieces can be identified with the vector (x1,...,xa) of the lengths of the pieces, listed from left to right, and since xi = 1, such a partition can be viewed as an element of ∆a−1. P Given d “cakes” C1,...,Cd, we consider partitions P of their union, the t-th cake 1 1 d d Ct being partitioned into at slices. Then P = ((P1 ,...,Pa1 ),..., (P1 ,...,Pad )) is d an element of P := t=1 ∆at−1. The subintervals of Ct in the partition P are t t denoted by Ij (P ), j ≤ at, when ordered from left to right. So, the length of Ij (P ) t Q is Pj . We denote by J the set of all vectors ~j = (j1,...,jd), jt ∈ [at]. For every vector ~ ~ j = (j1,...,jd) ∈ J , let v(j) = (ej1 ,ej2 ,...ejd ) be the vertex of P corresponding to ~j. Our notation will sometimes not distinguish between ~j and v(~j). There is a set of “agents” (or “players”). Given a partition P , we wish to allocate to each agent a d-tuple of slices, one from each cake. Such a d-tuple is determined ~ ~ t by a vertex v(j) of P (that is, by a vector j ∈ J ) — choosing the slice Ijt from Ct FRACTIONALLY BALANCED HYPERGRAPHS 3 for each t ∈ [d]. Of course, we want the d-tuples ~ja and ~jb of slices allocated to two different agents a,b to be disjoint, namely component-wise distinct. The agents are choosy. Each agent i has, for each partition P ∈ P, a list Li(P ) of acceptable d-tuples ~j of slices, indicated by their indices. So, for example, i L (P )= {(3, 5, 2), (1, 4, 2)} (here d =2, a1 = a2 = 3) means that agent i is ready 1 2 3 1 2 3 i to accept in the partition P either (I3 , I5 , I2 ) or (I1 , I4 , I2 ). Thus, L is a multi- valued function from P to J . Its inverse is denoted by Ai. Formally, For every vector ~j ∈ J and every agent i, i i −1 ~ ~ i A~j := (L ) (j)= {P ∈ P | j ∈ L (P )}. So Ai is the set of partitions in which agent i is ready to accept the d-tuple ~j. ~j The next observation expresses natural conditions on the lists Li(P ) as a KKM- cover condition on the sets Ai . It is folklore among cake-dividers, but we have not ~j found an explicit formulation of it that we can cite. The support supp(P ) of a point P ∈P is the minimal face of P containing P . Observation 1.4. For every i, the sets Ai , ~j ∈ V (P), form a KKM-cover of P if ~j and only if (1) they are closed, and (2) for every partition P ∈ J there exists ~j ∈ V (P) such that P ∈ Ai and ~j t Pjt > 0 for every t ∈ [d]. Proof. Suppose that (1) and (2) hold. Let σ be a face of P and let P ∈ σ. By (2), there exists ~j with P t > 0 for all t ∈ [d] and such that P ∈ Ai . Clearly, then, jt ~j ~ i v(j) is a vertex of P and hence of σ, showing P ∈ v∈V (σ) Av. Thus the collection (Ai , v ∈ V (P)) forms a KKM-cover of P. v S Conversely, suppose that the sets Ai form a KKM-cover of P. Let P ∈P and let ~j S = supp(P ). By (1.2), there exists a vertex v = v(~j) of S, where ~j = (j1,...,jd), with P ∈ Ai . Since v is a vertex in S, P t > 0 for all t ∈ [d], as required in (2). ~j jt Condition (2) is called the “hungry agents” assumption. We shall always assume it, as well as (1). Definition 1.5. A set Q of agents is called placatable if there exists a function q φ : Q → V (P) such that q∈Q Aφ(q) 6= ∅ and the supports supp(φ(q)) are disjoint. The condition means thatT it is possible to placate every player q ∈ Q. A partition q P ∈ q∈Q Aφ(q) yields a division of the cakes, in which if every q ∈ Q receives the slices defined by the vertex φ(q) then she is happy, since φ(q) ∈ Lq. We call such T an allocation “admissible”. Our aim is to prove the existence of large placatable sets. Given a partition P ∈P, the maximal size of a set of agents placatable by P is denoted by νD(P ). Remark 1.6. In the literature the lists Li are called preference lists, and then an admissible division is called envy-free or fair. We prefer the “placability” and “admissibility” terminology, since the requirement that agent i’s portion is in Li(P ) is not a preference, it is absolute. And there is no issue of envy or fairness - nobody squints at other agents’ portions. 4 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
The basic theorem in the field, Theorem 1.9 below, is that if d = 1 and the number of agents in Q is equal to the number of slices in the division, then Q is placatable. Meunier and Su [19] gave a new proof of this result. We use their approach to reduce the admissible division problem with d cakes to a problem on matchings in (d + 1)-uniform hypergraphs, where the reduction is one-directional: the existence of a matching in a hypergraph satisfying certain conditions implies the existence of large placatable sets. See Theorem 2.4 below for an exact formulation. ad Notation 1.7. We write (n; a1,a2,...,ad) −−→ m if for every instance of the admissible division problem with n agents and d cakes, where cake t is partitioned D into at parts, there exists a partition P ∈ t≤d ∆at−1 with ν (P ) ≥ m. Let ad ad (n; a1,a2,...,ad) be the largest number mQsuch that (n; a1,...,ad) −−→ m.
Note that n and the arguments at play different roles, and are not interchange- able. ad Remark 1.8. The relation (n; a1,a2,...,ad) −−→ m is monotone in all arguments ad ad (see Proposition 8.1). Namely, if (n; b1,b2,...,bd) −−→ m then (n; a1,a2,...,ad) −−→ m whenever a1 ≥ b1,...,ad ≥ bd. 1.3. Known results. The classical admissible division result, due to Stromquist [22] and Woodall [25], has already been mentioned. It is that if the number of pieces of a single cake is equal to the number of players, then there is an admissible partition. In our notation, this result says: ad Theorem 1.9. For all n ≥ 1: (n; n) −−−→ n, hence ad(n,n)= n. ad For d> 1 things are more complicated. For example, for (2;2, 2)¬ −−→ 2 (here “¬” stands for negation), as shown in [7]. Some other known results are: Theorem 1.10. ad ad • (2;2, 3) −−−→ 2 and (3;2, 2) −−−→ 2 [7]. ad • (3;5, 5) −−−→ 3 [16]. ad p • For d ≥ 2, (p; n,...,n) −−−→ 2d(d−1) whenever p ≤ d(n − 1)+1, and d times ad • (p; n,...,n) −−−→ p if p divides d(n − 1) + 1 [20]. | {zd(d}−1) d times ad • In particular,| {z } (2n − 1; n,n) −−−→ n.
1.4. Summary of our results. The simplest and possibly most attractive case ad of the −−→ relation is that of (n; n, ∗), namely when there are n agents, and one of the two cakes is partitioned into n slices. One question is then into how many slices should the second cake be partitioned in order to make all n agents content ad — what p>n guarantees (n; n,p) −−→ n. We show that p = n2 − n/2 suffices, and ad p =2n − 2 does not. We also prove results of the form (n; n,p) −−→ m, for various values of p. We also give lower bounds on ad(n; n, 2n − 1), but they fall short of ad determining whether (n; n, 2n − 1) −−→ n. ad Here is a complete list of the results we prove on the −−→ relation. FRACTIONALLY BALANCED HYPERGRAPHS 5
ad ⌊2r⌋n ad 2rn • (n; n,rn) −−→ ⌊2r⌋+2 and (n; n,rn) −−→ ⌈2r⌉+2 for every r ≥ 1 such that rn is an integer.l Inm particular: l m ad ad – (n; n,n2 − n/2) −−→ n (generalizing the result (2, 2, 3) −−→ 2 from [7]). n ad – (n; n, 2 ) −−→ n − 1. ad 2n−1 3n – (n; n, 2n − 1) −−→ max 3 , 5 . ad ad • (n;2n − 1, 2n − 1) −−→ n (generalizing the result (3; 5, 5) −−→ 3 from [16]). ad ad • (2n−2; n,n)¬ −−→ n (witnessing sharpness of the result (2n−1; n,n) −−→ n from [20]). ad ad • (n; n, 2n − 2)¬ −−→ n (generalizing the result (2; 2, 2)¬ −−→ 2 from [7]). As mentioned above, we shall reduce (in a one-way sense) problems on the ad bm “−−→” relation to a relation involving matchings, that will be denoted “−−−→” bm bm as in: (n,a1,...,ad) −−−→ m. Theorem 2.4 states that the −−−→ relation is stronger ad bm ad than the −−→ relation: if (n,a1,...,ad) −−−→ m then (n; a1,...,ad) −−→ m. Most bm of the results above will be proved via the corresponding result on the −−−→ relation. bm We shall also study the −−−→ relation on its own. We next define it.
2. Fractionally balanced hypergraphs and admissible division 2.1. Fractionally-balanced hypergraphs. A hypergraph will be identified with its edge set. A matching in a hypergraph H is a set of disjoint edges. The matching number ν(H) is the largest size of a matching in H. A fractional matching in H is a non-negative function f on H such that for every vertex v,
degf (v) := f(e) ≤ 1. e∈XH,e∋v ∗ The fractional matching number ν (H) is the maximum of |f| := e∈H f(e) over all fractional matchings f of H. P A hypergraph H is d-partite if its vertex set can be partitioned as V1 ∪···∪ Vd, in such a way that |e ∩ Vt| = 1 for every e ∈ H and t ∈ [d]. We are tacitly assuming that the partition is given and fixed, even though there may be more than one partition satisfying the condition. The sets Vt are called the sides of H. Definition 2.1 (Fractionally balanced hypergraphs). A function f : H → R+ is said to be balanced if it has constant degrees on every side Vt, namely degf (v) = degf (u) whenever u, v ∈ Vt. A hypergraph H is (a1,...,ad)-fractionally balanced if H is d-partite with respective side sizes a1,...,ad, and there exists a balanced + non-zero function f : H → R . Moreover H is (a1,...,ad)-fractionally uniformly balanced if for every vertex in V1, its neighbourhood, viewed as a (d − 1)-partite hypergraph, is the same.
Observation 2.2. If H is an (a1,...,ad)-fractionally balanced hypergraph then ∗ ν (H) = mint∈[d] at. bm Notation 2.3. We write (a1,a2,...,ad) −−−→ m if every (a1,a2,...,ad)-fractionally balanced d-partite hypergraph contains a matching of size m. Let bm(a1,a2,...,ad) bm be the largest number m such that (a1,a2,...,ad) −−−→ m. 6 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
bm K˝onig [13] proved that (n,n) −−−→ n for all n ≥ 1. ad We do not know if Remark 1.8 regarding the monotonicity of the “−−→” relation bm applies also to the “−−−→” relation. In fact, we suspect it is not (see Section 8 below).
2.2. Reducing admissible division to fractionally-balanced hypergraph bm matching. The relevance of −−−→ to admissible division stems from the follow- ing theorem: bm Theorem 2.4. For all (d + 1)-tuples (n,a1,...,ad), if (n,a1,...,ad) −−−→ m then ad (n; a1,...,ad) −−−→ m. In other words, ad(n,a1,...,ad) ≥ bm(n,a1,...,ad). This theorem was proved in Meunier and Su [19] for d = 1, and their method of proof, that uses triangulations, can be applied to prove also the general case. We give an alternative proof, that uses a generalization of the KKM theorem due to Komiya [15]. We shall need a slightly generalized version of the latter. Theorem 2.5 (Komiya [15]). Let R be the product X ×Y of two polytopes. Suppose that for every nonempty face α = σ × τ of R (σ a face of X, τ a face of Y ) there is given a set Bα = Vα × Wα, where Vα is an open set in X and Wα is a closed set in Y , satisfying the condition:
(2.1) σ ⊆ Bτ for every face σ of R. τ[⊆σ Suppose further that in each face α of R there is a point yα chosen, in partic- ular a point yR ∈ R. Then there exist faces σ1,...,σm of R such that yR ∈ m conv{yσ1 ,...,yσm } and i=1 Bσi 6= ∅.
The original theoremT is formulated with Bα closed (for the pedantic - with X = ∆0). The general version is obtained using a standard technique, of replacing each Vα by a closed subset of it, while maintaining the intersection pattern of the sets Vα.
Proof of Theorem 2.4. Let D be a copy of ∆n−1, and let R = D ×P, where, as before, P = ∆a1−1 × ... × ∆ad−1. Let V (D)= {u1,...,un}. For every vertex v = (u ,~j) of R, let B = ⋆(u ) × Ai , where ⋆(u) is the set of i v k ~j points in D having u in their supports. Note that Ai is closed and ⋆(u) is open. ~j For all other faces σ of R (namely faces with positive dimension) let Bσ = ∅. For every face σ of R let yσ be the barycenter of σ.
Claim 2.6. The sets Bv satisfy (2.1). To see this, let σ = γ × δ be a face of R (γ a face of D, δ a face of P) and let w = (~x, P ) be a point in σ, where ~x ∈ ∆n−1 and P ∈ P. We have to show that w ∈ Bv for some vertex v ∈ V (σ). There exists i for which xi 6= 0. Since the sets Ai form a KKM-cover, for some vertex ~j ∈ V (δ) we have P ∈ Ai . Then w ∈ B ~j ~j v for v = (ui,~j) ∈ V (σ). By Theorem 2.5 it follows that there exists a set Z ⊂ V (R)anda point(~x, P ) ∈ R such that
(1) (~x, P ) ∈ v∈Z Av, and (2) conv(Z) contains the barycenter of R. T FRACTIONALLY BALANCED HYPERGRAPHS 7
Let H be a (d + 1)-partite hypergraph, in which the first vertex side is V (D), and the q-th side, q > 1, is V (∆aq −1), and whose edge set is the elements of Z. Condition (2) above means that some convex combination of the edges of H gives 1 on all vertices of D, and 1 on all vertices of side q, q> 1, meaning that H is n aq −1 fractionally balanced. By the condition of the theorem it follows that H contains a matching M = {h1,...,hm} of size m, where hi = (ui,~ji) is an edge of H (so ui ∈ V (D) and i P ∈ A ). Then allocating to agent ui the pieces defined by ~ji is an admissible ~ji assignment for m agents, as required. Remark 2.7. A different approach for proving results on admissible division is the owner assignment method, initiated by Simmons and Su [23]. It is based on the observation that any (n − 1)-dimensional polytope has a triangulation T that is as fine as we wish, and an assignment of integers in [n] to the vertices of T , such that the vertices of every simplex of T are assigned distinct integers. This approach was used, for example, in [21]. Recently, Nyman et al. [20] used the owner-assignment approach to prove that ad (2n − 1; n,n) −−→ n. This result does not follow from Theorem 2.4, since as we bm shall see, (2n − 1,n,n)¬ −−−→ n. This will also show that the converse of Theorem 2.4 is false and bm(2n−1,n,n) < ad(2n−1,n,n). On the other hand, we shall meet corollaries of Theorem 2.4 that do not follow directly from the owner assignment method. Remark 2.8. By the symmetry of the function bm, Theorem 2.4 implies:
bm(a1,a2,a3) ≤ min ad(a1; a2,a3), ad(a2; a3,a1), ad(a3; a1,a2) We suspect that there are cases of strict inequality, but know no examples witnessing that.
3. Yet another topological tool To the topological tools introduced so far we add another – a topological gener- alization of Hall’s marriage theorem. A hypergraph C is called a simplicial complex if it is closed down, namely e ∈ C, f ⊆ e imply f ∈ C. For a subset X of V (C) let C[X] = {e ∈ C | e ⊆ X}. C is called homologically k-connected if for every −1 ≤ j ≤ k, the j-th reduced simplicial homology group of C with rational coefficients H˜j (C) vanishes. It is called homotopically k-connected if for every −1 ≤ j ≤ k any continuous function from Sj to the geometric realization ||C|| of C can be extended continuously to Bj+1. The homological (resp. homotopic) connectivity η(C) (resp. ηh(C)) of C is the largest k for which C is homologically (resp. homotopically) k-connected, plus 2. It is known that η ≥ ηh, with equality if ηh ≥ 3. Given sets V1,...,Vn and a set K ⊆ {1,...,n}, let VK = i∈K Vi. Given a simplicial complex C and not necessarily disjoint subsets Vi of V (C), a C-transversal S is a set in C formed by a not necessarily injective choice of one vertex from each Vi.
Theorem 3.1 (Topological Hall Theorem). If η(C[VK ]) ≥ |K| for every K ⊆ [n] then there exists a C-transversal. 8 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
By the above remark, this implies the homotopic version, with ηh replacing η. In the discussion below, η can be replaced throughout by ηh. The homotopic version of the theorem is essentially proved (but not explicitly stated) in [4]. The explicit homotopic version of the theorem was noted by Aharoni (see remark following Theorem 1.3 in [17]). In the homological form it was stated and proved explicitly as Proposition 1.6 in [18]. A standard argument of adding dummy vertices yields a deficiency version of Theorem 3.1:
Theorem 3.2. Let C be a simplicial complex, V1,...,Vn subsets of V (C), and d ≥ 0 an integer. If η(C[VK ]) ≥ |K|− d for every K ⊆ [n], then the system has a partial C-transversal of size n − d. We shall apply these theorems to independence complexes of graphs. The in- dependence complex I(G) of a hypergraph G consists of the independent sets in V (G) (a set is independent if it does not contain an edge of G). The line graph L(G) of a graph G has E(G) as its vertex set, and two edges in E(G) form an edge in L(G) if they intersect. Clearly, an independent set in L(G) is a matching in G, so I(L(G)) is the matching complex of G, usually denoted M(G). A hypergraph H is called bipartite with sides X, Y (playing a-symmetrical roles) if V (H) = X ∪ Y , X ∩ Y = ∅ and |e ∩ X| = 1 for all e ∈ H. We shall apply Theorems 3.1 and 3.2 to such hypergraphs. For every x ∈ X let NH (x) be the neighborhood of x in Y , namely {f ⊆ Y | f ∪{x} ∈ H}. For a subset K of X, let
NH (K) := x∈K N(x). The means that we treat NH (K) as a multi-set (and a multi-hypergraph): identical neighbors of two elements of K induce two elements U U in NH (K). If H is a d-partite hypergraph, then NH (K) is an (d − 1)-partite multi- hypergraph. Applied to this setting, Theorem 3.2 yields: Corollary 3.3. Let H be a bipartite hypergraph with sides X, Y and d ≥ 0 an integer. If η(M(NH (K))) ≥ |K|− d for every K ⊆ X, then H has a matching of size |X|− d. In order to apply Theorems 3.1, 3.2, 3.3, one needs combinatorially formulated lower bounds on η, in particular on η(I(G)) for a graph G. A general lower bound on η(I(G)) is due to Meshulam [18]. Given an edge e in a graph G we denote by G − e the graph obtained by removing e, and by G¬e the graph obtained by removing the vertices of e and all their neighbors (together with the edges incident to them). The Meshulam bound is given by: Theorem 3.4. For every edge e in a graph G η(I(G)) ≥ min(η(I(G − e)), η(I(G¬e)) + 1).
In [3] this was proved also for ηh. This bound is conveniently expressed in terms of a game between two agents, CON (wishing to prove high connectivity) and NON (the Mephistophelian “spirit of perpetual negation”), on the graph G. At each step, CON chooses an edge e from the graph remaining at this stage, where in the first step the graph is G. NON can then either (1) delete e from the graph (we call such a step a “deletion” or “disconnection”), or FRACTIONALLY BALANCED HYPERGRAPHS 9
(2) remove , from the graph the two endpoints of e, together with all neighbors of these vertices and the edges incident to them (we call such a step an “explosion”, and denote by G¬e the resulting graph). The result of the game (payoff to CON) is defined as follows: if at some point there remains an isolated vertex, the result is ∞ (when there is an isolated vertex v, the independence complex is contractible to v, hence it is infinitely connected). Otherwise, at some point all vertices have disappeared, in which case the result of the game is the number of explosion steps. We define Ψ(G) as the value of the game, i.e., the result obtained by optimal play on the graph G. Convention 3.5. Henceforth we shall assume that NON always chooses the best strategy for him, namely he removes e if min(Ψ(G − e), Ψ(G¬e)+1)= Ψ(G − e), and explodes it if min(Ψ(G − e), Ψ(G¬e)+1)=Ψ(G¬e) + 1. Theorem 3.4 can be stated as: Theorem 3.6. η(I(G)) ≥ Ψ(G). The game formulation first appeared in [5]. For an explicit proof of Theorem 3.6 using the recursive definition of Ψ, see Theorem 1 in [1].
4. Lower bounds on bm and on ad In this section we prove lower bounds on values of bm, and thereby on values of ad. A classical result in this direction is F¨uredi’s theorem [10], that states that in a d-partite hypergraph ν ≥⌈ν∗/(d − 1)⌉. By Observation 2.2 this entails:
Theorem 4.1. For any integers a1,...,ad, bm (a1,...,ad) −−−→ min at (d − 1) . t≤d In particular we have:
bm n Theorem 4.2. If m ≥ n, then (n,n,m) −−−→⌈ 2 ⌉. The Fano plane minus a point (the truncated projective plane of order 2) is called the Pasch hypergraph. It is a (2, 2, 2)-fractionally-balanced tripartite hypergraph n with a maximum matching of size 1. Taking 2 vertex-disjoint copies of it (when n n is even), or 2 copies plus one isolated edge (when n is odd), yields an (n,n,n)- fractionally-balanced tripartite graph in which the maximum matching size is ⌈n/2⌉. So we have: bm n Theorem 4.3. (n,n,n)= 2 for all n ≥ 1. For further results, we will use the following lemma: Lemma 4.4. Let G be a bipartite graph with sides B, C and let s ≥ 1 be a real number. Let f be an integral weight function on E(G) with the following properties: • f(b,c) ∈{0, 1, 2} for all (b,c) ∈ E; • degf (b) ≤ 2s for all b ∈ B; • degf (c) ≤ 2 for all c ∈ C. Then |f| η(M(G)) ≥ . 2s +2 10 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
Proof. Since M(G) = I(L(G)), by Theorem 3.6 it is sufficient to prove that |f| Ψ(L(G)) ≥ 2s+2 . This can be proved by playing Meshulam’s game on L(G). One canl view mG as an |B|-by-|C| array. Each edge (b,c) ∈ G corresponds to a cell in row b ∈ B and column c ∈ C. In L(G), the vertices are the cells, and each edge corresponds to a pair of cells in the same row or the same column. By assumption, the total weight in each row is at most 2s, and in each column at most 2. We show that, if CON offers pairs of cells in a specific order, then each explosion made by NON destroys cells with a total weight of at most 2s + 2. This implies |f| that NON needs at least 2s+2 explosions to destroy all edges. CON starts by offeringl pairsm (b,c1), (b,c2) of cells in the same row such that f(b,c1)+ f(b,c2) ≥ 2. These are the pairs in which the weight of at least one cell is 2, or the weight of both cells is 1. If NON explodes such a pair, then one row and two columns are destroyed. Since a weight of at least 2 is common to the row and columns, the total weight destroyed is at most (2s +2+2) − 2=2s + 2. If NON disconnects all such pairs, then CON goes on to offer all pairs of cells in the same column (b1,c), (b2,c) such that f(b1,c) = f(b2,c) = 1. Each such cell is now connected, in its row, only to cells with weight 0. Therefore, if NON explodes such a pair, then the total destroyed weight is the weight in column c, which is 2 < 2s + 2. If NON disconnects all offered pairs, then CON offers all pairs of cells in the same row (b,c1), (b,c2) such that f(b,c1) = 1 and f(b,c2) = 0. The cell (b,c1) is connected, in its column, only to cells with weight 0. Therefore, an explosion destroys only the weight in row b which is at most 2t, and the weight in column c2 which is at most 2. Finally, CON offers all remaining connected pairs of cells. Each cell is now connected, in its row, only to cells with weight 0. Therefore, an explosion destroys only the weight in at most two columns, which is at most 2.
Corollary 4.5. Let G be a bipartite graph with sides B, C, let n ≥ 1 be an integer, and let r ≥ 1 be a real number. If there exists a function f : E(G) → R+ such that degf (b) ≤ 1 for all b ∈ B and degf (c) ≤ 1/r for all c ∈ C, then ⌊⌊2r⌋ · |f|⌋ (a) η(M(G)) ≥ . ⌊2r⌋ +2 ⌊2r · |f|⌋ (b) η(M(G)) ≥ . ⌈2r⌉ +2 Proof. (a) Let P be the polytope consisting of those vectors ~x ∈ RE(G) satisfying the following constraints: (1) ~x ≥ ~0
(2) b∈e x(e) ≤⌊2r⌋ for all b ∈ B, (3) c∈e x(e) ≤ 2 for all c ∈ C, and (4) P x(e) ≥ ⌊⌊2r⌋ · |f|⌋. Pe∈E Then ⌊2Pr⌋· f ∈ P , so P is nonempty. Since G is bipartite and all constraints are integral, P is integral. Hence, there exists an integer vector f ∗ in P . Constraints (1) and (3) imply that all coordinates of f ∗ are in {0, 1, 2}. Then (4) implies FRACTIONALLY BALANCED HYPERGRAPHS 11
|f ∗|≥⌊⌊2r⌋ · |f|⌋ , and (2) and (3) imply that f ∗ satisfies the conditions of Lemma 4.4 for 2s = ⌊2r⌋. (b) Suppose now that P satisfies the following constraints: (1) ~x ≥ ~0
(2) b∈e x(e) ≤⌈2r⌉ for all b ∈ B, (3) c∈e x(e) ≤ 2 for all c ∈ C, and (4) P x(e) ≥⌊2r · |f|⌋. Pe∈E Then 2rP· f ∈ P , and all constraints are still integral, so P is integral and contains an integer vector f ∗ as above. Now |f ∗|≥⌊2r · |f|⌋, and f ∗ satisfies the conditions of Lemma 4.4 for 2s = ⌈2r⌉.
bm We shall use the lemma to prove “−−−→” relations.
Theorem 4.6. For every r ≥ 1 such that rn is an integer:
bm ⌊2r⌋ n (a)[n,n,rn] −−−→ ⌊2r⌋ +2 bm 2rn (b)[n,n,rn] −−−→ ⌈2r⌉ +2 Proof. Let H be the fractionally balanced hypergraph with sides A, B, C with |A| = |B| = n and |C| = rn, and let f be a corresponding weight function on H, with degf (a) = degf (b)=1 for a ∈ A, b ∈ B and degf (c)=1/r for c ∈ C. For any K ⊆ A, the fractional matching induced by f on NH (K) satisfies the conditions of Corollary 4.5 with |f| = |K|. By part (a) of the corollary we have
⌊2r⌋ |K| η(M(N (K))) ≥ H ⌊2r⌋ +2 ⌊2r⌋ n ≥ |K|− n − since n ≥ |K|. ⌊2r⌋ +2
⌊2r⌋n Corollary 3.3 with d = n − ⌊2r⌋+2 implies that H has a matching of size at least ⌊2r⌋n l m n − d = ⌊2r⌋+2 . Similarly,l bym part (b) of Corollary 3.3 we have
⌊2r|K|⌋ η(M(N (K))) ≥ H ⌈2r⌉ +2 ⌊2rn⌋ ≥ |K|− n − since n ≥ |K|. ⌈2r⌉ +2
⌊2rn⌋ Corollary 3.3 with d = n − ⌈2r⌉+2 implies that H has a matching of size at least ⌊2rn⌋ 2rnl m n − d = ⌈2r⌉+2 = ⌈2r⌉+2 (the last equality is true since 2rn is an integer), proving partl (b)m of thel theorem.m 12 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
Corollary 4.7. For all n ≥ 2: bm (1) [n,n,n2 − n/2] −−−→ n n bm (2) [n, n, ] −−−→ n − 1 2 bm 2n − 1 3n (3) [n, n, 2n − 1] −−−→ max , 3 5 Proof. (1) Apply Theorem 4.6(a) with r = n − 1/2. Then ⌊2r⌋ = 2n − 1 and ⌊2r⌋n n(2n−1) 2n ⌊2r⌋+2 = 2n+1 = n − 2n+1 = n. l m l m l m ⌊2r⌋n (2) Apply Theorem 4.6(a) with r = (n−1)/2. Then ⌊2r⌋ = n−1, and ⌊2r⌋+2 = (n−1)n l m n+1 = n − 1. l m ⌊2r⌋n 3n (3) Apply Theorem 4.6 with r =2 − 1/n. Then part (a) gives ⌊2r⌋+2 = 5 , 2rn 4n−2 2n−1 l m and part (b) gives ⌈2r⌉+2 = 6 = 3 . l m ad As already mentioned, Cloutier et al. [7] proved that (3, 5, 5) −−→ 3. Here we strengthen and generalize this result as follows:
bm n Theorem 4.8. If k ≤ n then (k,n,n) ≥ min(k, 2 ) Proof. Let H be a fractionally balanced hypergraph with sides A, B, C, where |A| = n k and |B| = |C| = n. Let d := k − min(k, 2 ). Recall that, for any K ⊆ A, NH (K) is its set of neighbors in H (so, it is a subset of the bipartite multigraph B × C). By Corollary 3.3 it suffices to show
(4.1) η(M(NH (K))) ≥ |K|− d. for every subset K of A. Let f be a function on H witnessing its fractional balanced- n ness, with degf (a)= k for every a ∈ A and degf (z) = 1 for every z ∈ B ∪ C. Let g be f restricted to K and induced on B × C, so g(b,c)= a∈K:(a,b,c)∈H f(a,b,c). n|K| Then |g| = k , and degg(z) ≤ 1 for every z ∈ B ∪ C. TheP function g satisfies the n conditions of Corollary 4.5 with r = 1 and |g| = k |K|. Therefore, by either part of Corollary 4.5: ⌊2 · |g|⌋ ⌊2n · |K|/k⌋ η(M(N (K))) ≥ = H 4 4 We claim that the right-hand side is at least |K|− d. Standard manipulations of 2k−n 3 ceilings and floors yield the following equivalent statement: 2k |K|≤ d + 4 . The right-hand side is positive. Hence, if the left-hand side is negative then we are done; otherwise, since |K|≤ k, it is sufficient to prove the claim for |K| = k, which n 3 n 3 n yields k − 2 ≤ d + 4 . This is equivalent to 2 + 4 ≥ min(k, 2 ), which is true. Another route can be taken, using the fact that g is a fractional matching in ∗ n|K| NH (K), implying ν(NH (K)) = ν (NH (K)) ≥ |g| = k . In [4] it was proved (not ν(G) stated explicitly) that η(M(G)) ≥ 2 in any graph G (this can also be proved FRACTIONALLY BALANCED HYPERGRAPHS 13 easily using the Meshulam game). Since η is integral, this implies n|K| (4.2) η(M(N (K))) ≥ . H 2k n|K| ⌊2n·|K|/k⌋ Claim (4.1) now follows by the previous argument, since 2k ≥ 4 . bm Corollary 4.9. (n, 2n − 1, 2n − 1) −−−→ n.
5. Extending the arrows relations to hypergraphs of higher uniformities
We prove a result on adding a coordinate, assuming it is large enough, for the bm “−−−→” relation: bm bm Theorem 5.1. If (a1,...,ad) −−−→ m then (a1,...,ad,ad+1) −−−→ m for some ad+1. The main tool we shall use in the proof of Theorem 5.1 is the so-called “Gordan’s lemma” from convex geometry. Recall the dual of a convex cone X ⊂ Rk is X∗ := {v ∈ Rk : hx, vi≥ 0 ∀x ∈ X}. A polyhedral cone in Rk is said to be rational if its extreme rays are multiples of vectors with rational coordinates. Theorem 5.2 (Gordan’s Lemma). If X ⊂ Rk is a rational convex polyhedral cone, then the semigroup X∗ ∩ Zk, with the operation of coordinate-wise addition, is finitely generated. Equivalently: if A is an integral matrix, then there is a finite set S of vectors in C = {~x | A~x ≥ ~0} ∩ Zk such that every vector in C is a combination of vectors from S with integral coefficients. Fix a1,...,ad. Denote by BH(a1,...,ad) the collection of d-partite (a1,...,ad)- [a ] RQt t fractionally balanced hypergraphs, and by W (a1,...,ad) ⊂ + the collection of all possible balanced weight functions on a complete d-partite hypergraph with side sizes a1 ...,ad. Moreover, if H ∈ BH(a1,...,ad), then let
WH := {w ∈ W (a1,...ad) : supp w ⊂ H} be the collection of weight functions on H witnessing its balanced-ness. Note that W (a1,...,ad), WH are defined by the intersection of a finite collection of closed half-spaces, such that the inequalities defining them have integer coefficients. By definition, this means that they are rational convex polyhedral cones. Hence we have:
Q [at] Claim 5.3. W (a1,...,ad) ∩ Z t is finitely generated as a semigroup. Example 5.4. • The n! characteristic functions of perfect matchings form a Gordan base for W (n,n). To see this, note that if f is an integral function on E(Kn,n) with constant degrees, then by Hall’s theorem supp(f) contains a perfect match- ing F . By an induction hypothesis g := f − χF is a sum of characteristic functions of perfect matchings, and hence so is f = g + χF . • A similar argument shows that W (n,sn) is generated by the characteristic functions of unions of n vertex-disjoint stars K1,s. 14 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
Proof of Theorem 5.1. Choose ad+1 = N sufficiently large (to be determined be- low). [a1]×···×[ad]×[N] ′ ′ R Suppose H ∈ BH(a1,...,ad,N). Then the cone WH ⊂ + is rational and nonzero, hence it contains a nonzero integral point w′. d [a ] RQt=1 t d Define w ∈ + as follows. For each e ∈ t=1[at], let w(e) := w′(e,Q j). jX∈[N] ′ Since w is balanced, w is also balanced. Specifically, w ∈ WH , where H is the d-partite hypergraph obtained from H′ by removing the (d + 1)-st vertex side [N]. ′ Now, as w is balanced, every vertex in the t-th class [at] has the same total ′ w -weight, say bt. By double-counting, atbt = ad+1bd+1 = Nbd+1 for any t ≤ d. Let U be the finite generating set for W (a1,...,ad) given by Claim 5.3. Let b = max {u(e) | e ∋ v}. v∈V (H),u∈U X Since w ∈ WH ⊂ W (a1,...,ad), there is a decomposition T w = uℓ Xℓ=1 for some u1,...,uT ∈ U. Here, if a function u appears in the sum with coefficient c, we decompose it as a sum of c copies of u. Since every vertex in [at] has w-weight bt, it follows that bt ≤ bT . Let Hℓ ∈ BH(a1,...,ad) be the support of uℓ for each ℓ. By assumption, Hℓ contains an m-matching Mℓ. The number of different possible such Mj is
d a q = t · m!d−1. m t=1 Y So provided N ≥ bq · max a , it follows T ≥ bt = Nbd+1 ≥ qb , and the pigeon- t t b atb d+1 hole principle shows that some bd+1 different Mℓs are identical. Without loss of generality M1 = ··· = Mbd+1 = M. So uℓ(e) ≥ 1 for every ℓ ≤ bd+1 and every edge e ∈ M. Let E ⊂ M. Write J for the set of vertices j in the (d +1)-st part of H′ having w′(e, j) ≥ 1 for some e ∈ E. Such (e, j) are necessarily edges of H′. Then:
bd+1 ′ bd+1|E|≤ uℓ(e) ≤ w(e)= w (e, j) Xℓ=1 eX∈E eX∈E eX∈E jX∈[N] ′ = w (e, j) ≤ |J|bd+1. Xj∈J eX∈E By Hall’s theorem this implies that there is a injection g : M → [N] such that (e,g(e)) ∈ H for every e ∈ M, yielding an extension of M to a (d + 1)-partite matching of size m in H′.
In Alon and Berman [6] a geometric proof of Gordan’s lemma was given, pro- viding an explicit bound. This can be used to give an upper bound on N, but we shall not pursue this. FRACTIONALLY BALANCED HYPERGRAPHS 15
k Question 5.5. Does there exist N = O( t=1 at) satisfying the conclusion of Theo- rem 5.1? Q Also, given that the converse of Theorem 2.4 is false, we do not know whether ad Theorem 5.1 is true for −−→. ad Conjecture 5.6. If (a1; ...,ad) −−−→ m then there exists ad+1 such that ad (a1; ...,ad,ad+1) −−−→ m.
6. Upper bounds on ad and bm In this section we prove non-existence results for hypergraph matchings (the bm ad relation −−−→) and admissible division (the relation −−→), implying upper bounds on the functions ad and bm. 6.1. Upper bounds for hypergraph matchings. Theorem 6.1. Suppose k
• H1 = {aibici | i ∈ I}∪{aibm+icm+i | i ∈ I}, • H2 = {aj bici+m | i ∈ I, j ∈ J}∪{ajbi+mci | i ∈ I, j ∈ J}, • H3 = {aj bncn | j ∈ J}. 1 1 If n is even, let H = H1 ∪ H2. Assigning weights 2 to every edge in H1 and n to every edge in H2, we get deg ai = 1 and deg bi = deg ci = k/n, so H is fractionally 3n balanced. We claim that the largest matching in H is of size 4 . Indeed, let M be a maximum matching in H with x edges in H1. Then M can contain at most min{k − m, 2(m − x)} edges in H2. So k + m 3n ν(H) ≤ x + min{k − m, 2(m − x)} = min{k − m + x, 2m − x}≤ ≤ . 2 4 3 If n is odd, let H = i=1 Hi. To see that H is fractionally balanced, assign weights 1 to every edge in H , 1 − k to every edge in H , and k 2 S 1 n−1 n(n−1)(k−m) 2 n(k−m) to every edge of H3, to get again deg ai = 1 and deg bi = deg ci = k/n. H3 adds at most one edge to the maximum matching. Theorem 6.1 can be improved in some special cases: n bm n Theorem 6.2. Write m = ⌊ 2 ⌋. If k−m divides m then (k,n,n) ≤ min(k, 2 ). Proof. Write k′ = k − m. Since k − m divides m we have m = qk′ for some integer q ≥ 1. Let I,J,H1,H3 be as in the proof of Theorem 6.1. For every 1 ≤ i ≤ q define i ′ H4 ={aj+mb(i−1)k′ +jc(i−1)k′ +j+m | 1 ≤ j ≤ k } ′ ∪{aj+mb(i−1)k′+j+mc(i−1)k′+j | 1 ≤ j ≤ k }, q i and let H4 = i=1 H4. S 16 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
If n is even, define H = H1 ∪ H4. Note that if M is a maximum matching in H containing x edges in H1, then it can contain at most m − x edges in H4, so n 1 ν(H) ≤ m = 2 . To see that H is fractionally balanced, assign weight 2 to every k−m edge of H1 and 2m to every edge of H4. n If n is odd, let H = H1 ∪ H3 ∪ H4. Now ν(H) ≤ m +1= ⌈ 2 ⌉. Assigning weight 1 k k−m k 2 to every edge in H1, and n(k−m) to every edge of H3, and n−1 − n(n−1) to every edge of H4, we obtain that H is fractionally balanced. Theorem 6.2 implies bm(n, 2n − 2, 2n − 2) ≤ n − 1, while Theorem 4.9 implies bm(n, 2n − 1, 2n − 1) ≥ n. Question 6.3. What is the value of bm(n, 2n − 2, 2n − 1)? We do not even know what is bm(3, 4, 5). 2rn Theorem 6.4. Let r ≥ 1 be such that both rn, 2r+1 are integers. Then for any 2rn bm 2rn bm 2rn 2r+1 ≤ k ≤ rn, (n,n,k)¬ −−−→ 2r+1 +1, implying (n,n,k) ≤ 2r+1 for such k. 2rn Proof. Let N be such that k = N + 2r+1 . For such N the system of equations
2rn Ny + 2r+1 x =2r − 1
2rn 2rn 2+ 2r+1 x = 2r+1 y have a non-negative solution in x and y: x = 2r+1 − 2r+1 ,y = 2r+1 . k rn k We construct a 3-partite hypergraph H. Let V (H)= A ∪ B ∪ C, where A and 2rn B are copies of [n], and C = [N + 2r+1 ]. Define f on A × B × C as follows: 2rn (1) f((i,i,j)) = y for i ∈ 2r+1 , j ∈ [N]. 2rn i h i 2rn (2) f((i, 2r+1 + 2r ,N + i))= 1 for i ∈ 2r+1 . 2rn i h 2rn i (3) f(( 2r+1 + 2r ,i,N + i))= 1 for i ∈ 2r+1 . 2rn h i (4) f((i,i,N +j)) = x for i, j ∈ 2r+1 . Let H be the support of f. h i 2rn 2rn For i ∈ A ∩ [ 2r+1 ] we have degf (i)= N · y +1+ 2r+1 · x, and for a larger element 2rn 2rn 2r+1 + j of A we have degf ( 2r+1 + j)=2r. By the choice of x and y these are equal. The same is true for B symmetrically. 2rn For j ∈ C ∩ [N] we have degf (j) = 2r+1 y and for N + j ∈ C \ [N] we have 2rn degf (N + j)=2+ 2r+1 x. Again, these are equal. Thus f is balanced, and H is fractionally balanced. 2rn We claim that ν(H) ≤ 2r+1 . To see this, note that if M is a matching of size 2rn 2r+1 +1, then it must contain edges of types (2) and (3), and its trace on A × B 2rn i 2rn i should contain two edges of the form (i, 2r+1 + 2r ) and ( 2r+1 + 2r ,i), but both these edges can be completed to an edge of H only by adding to them the element N + i from C. n−1 Putting r = 2 and r = 2, respectively, yields: Corollary 6.5. bm n (1) (n, n, 2 ) ≤ n − 1, (2) bm(n, n, 2n − 1) ≤⌊4n/5⌋. FRACTIONALLY BALANCED HYPERGRAPHS 17
We conjecture that (1) is tight:
n bm Conjecture 6.6. If n ≥ 4 then (n, n, 2 + 1) −−−→ n. By Theorem 4.1, bm(n,n,...,n) ≥ 2. We conjecture that it is tight too:
Conjecture 6.7. bm(n,n ··· ,n)=2.
n times It would suffice to| show{z that} for every n there exists a fractionally balanced n-partite hypergraph with sides of size n, having ν = 2.
Theorem 6.8. Conjecture 6.7 is true for every n of the form n = q or n = q +1 or n = q + p − 2 or n = q + k − 1, where q,p are integers for which q-uniform and p-uniform projective planes exist, and k ≤ log2 n. Proof. The constructions are disjoint unions of two of the following intersecting hypergraphs.
• Hq is the q-partite truncated projective plane, obtained from the q-uniform projective plane by removing a vertex and all the edges containing it. So Hq is a fractionally balanced q-partite hypergraph with sides of size q − 1 and a maximum matching of size 1. m • For an integer m ≥ 1, let Hq be the hypergraph obtained from Hq by m duplicating one vertex side m more times. So Hq is a fractionally balanced (q+m)-partite hypergraph with sides of size q−1 and a maximum matching of size 1. • In consists of a single n-sized edge {x1,...,xn}, where xi, i ≤ n are distinct vertices. • Jn consists of all n edges of the form {y1,...,yi−1, xi,yi+1,...,yn}, 1 ≤ i ≤ n, where yi, i ≤ n are distinct vertices. Note that In ∪ Jn is an intersecting fractionally balanced n-partite hypergraph with sides of size 2. • Let M be an n × k matrix whose entries are nk distinct vertices. The rows i of M will be the vertex sides of the n-partite hypergraphs Jn and Jn(k) we construct. Let M1,...,Mk be the columns of M. i For 1 ≤ i ≤ k let Jn be the hypergraph on the entries of M containing all the edges e satisfying the following three conditions: – e intersects every row of M exactly once, t−1 – |e ∩ Mt| =2 for every 1 ≤ t ≤ i − 1, and i−1 – |e ∩ Mi| = n − 2 + 1. k i k−1 Let Jn(k)= i=1 Jn. If k is small enough so that 2 − 1 < n/2, then Jn(k) is an intersecting fractionally balanced n-partite hypergraph with sides of size k. S Then the following are fractionally balanced n-partite hypergraphs with sides of size n and with ν = 2:
(1) Hq ∪ Iq, for n = q. 1 (2) Hq ∪ In ∪ Jn, for n = q + 1. p−2 q−2 (3) Hq ∪ Hp , for n = q + p − 2, p ≥ 3, q ≥ 3. k−1 (4) Hq ∪ Jn(k), for n = q + k − 1 ((1) and (2) are the cases k = 1 and k =2 of this construction). 18 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB
Assuming Goldbach’s conjecture (or even a slightly weaker conjecture that every even integer is the sum of two prime powers), (3) implies that the conjecture is true for all even n. 6.2. Bounds on the connectivity of matching complexes. As we have seen, bounds on the values of bm can be achieved through bounds on the connectivity of matching complexes. It is worth while studying such bounds on their own. Notation 6.9. For positive integers a,b let ζ(a,b) be the minimum of η(M(G)) over all (a,b)-fractionally balanced bipartite graphs G. Corollary 4.5 implies that, for every integer n ≥ 1 and number r ≥ 1 such that rn is an integer: ⌊2r⌋ n 2rn ζ(n,rn) ≥ max , . ⌊2r⌋ +2 ⌈2r⌉ +2 In particular: ζ(n,n2 − n/2) ≥ n n ζ(n, ) ≥ n − 1 2 2n − 1 3n ζ(n, 2n − 1) ≥ max , 3 5
Claim 6.10. ζ(n, (n − 1)2)
HD :={(i, j, j) | 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n} ∪{(i, j, j + 1) | n ≤ i ≤ 2n − 2, 1 ≤ j ≤ n}. FRACTIONALLY BALANCED HYPERGRAPHS 19
Note that HD is (2n − 2,n,n)-fractionally-balanced and has no matching of size n.
ad Theorem 6.11. For all n ≥ 2, (2n − 2; n,n)¬ −−−→ n; hence ad(2n − 2,n,n) Proof. We consider an instance of the cake-division problem with 2n − 2 agents, in which each cake is cut into n slices. We define for each agent i ∈ [2n − 2]: Ai := {(j, k) | (i, j, k) ∈ HD} A1 := {(j, j) | j ∈ [n]} if 1 ≤ i ≤ n − 1, = 2 (A := {(j, j + 1) | j ∈ [n]} if n ≤ i ≤ 2n − 2. Given partitions ~v, ~w of the two cakes, we define 1 1 B(~v, ~w) := {(j, k) | v ≥ , w ≥ }. j n − 1 k n − 1 The acceptable pairs of agent i are the pairs in B(~v, ~w) and the max-sum pairs in Ai: i ′ ′ L (~v, ~w) := B(~v, ~w) ∪ {(j, k) ∈ Ai : vj + wk ≥ vj′ + wk′ for all (j , k ) ∈ Ai}. First, we show closedness, namely, that for every i, j, k, the set Pi,j,k := {(~v, ~w) | i (j, k) ∈ L (~v, ~w)} is closed. If (j, k) 6∈ Ai, then Pi,j,k is the set {(~v, ~w) | vj ≥ 1 1 n−1 , wk ≥ n−1 }, which is closed since it is the intersection of the partition polytope 1 1 with two closed hyperspaces defined by vj ≥ n−1 and wk ≥ n−1 . If(j, k) ∈ Ai, then ′ ′ Pi,j,k is the union of the above set with {(~v, ~w) | vj +wk ≥ vj′ +wk′ for all (j , k ) ∈ Ai}, which is again defined by intersection of closed hyperspaces. Next, we show hungriness, namely, that for every i, ~v, ~w, the set Li(~v, ~w) contains at least one pair of nonempty slices. If B(~v, ~w) 6= ∅ then it obviously contains (only) pairs of nonempty slices. If B(~v, ~w) = ∅, then in at least one cake, say cake 1, all slices are shorter than 1/(n − 1). Since their total length is 1, all slices in that cake are nonempty. The set Ai contains n pairs, and the total length-sum of all pairs is 2. Therefore, the maximum length-sum of a pair is at least 2/n. Since 2/n ≥ 1/(n − 1), and all slices of cake 1 are shorter than 1/(n − 1), the slice of cake 2 in any pair maximizing the length-sum must be nonempty too. Finally, we prove that in every partition (~v, ~w), at most n − 1 agents can be allocated an acceptable pair. Case 1: every agent i gets a pair (ji, ki) ∈ Ai. Since the largest matching in {(i, ji, ki) | i ∈ [2n − 2]}⊂ HD is of size n − 1, at most n − 1 agents are satisfied. Case 2: at least one agent gets a pair from B(~v, ~w) \ Ai. The length-sum of this pair is at least 2/(n − 1). The length-sum of every max-sum pair in Ai is at 2 2 least 2/n. Hence, the length-sum of every n pairs is at least n−1 + (n − 1) · n = 2 2n+2(n−1) n2−n+1 n(n−1) = 2 n2−n > 2, which is a contradiction since the total length of the cakes is 2. ad Theorem 6.12. For all n ≥ 2, (n; n, 2n − 2)¬ −−−→ n; hence ad(n; n, 2n − 2) Note that this is different than Theorem 6.11, since the role of the first argument is different from that of the second and third arguments. 20 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB Proof. We consider a cake-cutting instance with n agents, in which cake #1 is cut into n slices and cake #2 is cut into 2n − 2 slices. We define for each agent i ∈ [n]: Ai := {(j, k) | (k,i,j) ∈ HD} (note the difference from Ai in Theorem 6.11) = {(i, k) | 1 ≤ k ≤ n − 1}∪{(i +1, k) | n ≤ k ≤ 2n − 2} Given partitions (~v, ~w), we define 1 B(~v, ~w) := {(j, k) | v ≥ , 1 ≤ k ≤ 2n − 2} j n − 1 The acceptable pairs of agent i are again the pairs in B(~v, ~w) and the max-sum pairs in Ai: i ′ ′ L (~v, ~w) :={(j, k) ∈ Ai : vj + wk ≥ vj′ + wk′ for all (j , k ) ∈ Ai} ′ ′ ∪{(j, k) ∈ B(~v, ~w): vj ≥ vj′ , wk ≥ wk′ for all (j , k ) ∈ B(~v, ~w)}. For the proof, we assume without loss of generality that w1 = max wk wn = max wk 1≤k≤n−1 n≤k≤2n−2 i so for all pairs in L (~v, ~w), the length of the second slice is either w1 or wn (for pairs in B(~v, ~w) it is max(w1, wn)). Closedness of the sets Pi,j,k can be proved similarly to Theorem 6.11: If (j, k) 6∈ 1 ′ ′ Ai, then Pi,j,k is the set {(~v, ~w) | vj ≥ n−1 , vj ≥ vj for all 1 ≤ j ≤ n, wk ≥ ′ wk′ for all 1 ≤ k ≤ 2n − 2}, which is closed since it is the intersection of the partition polytope with closed hyperspaces. If (j, k) ∈ Ai, then Pi,j,k is the union ′ ′ of the above set with {(~v, ~w) | vj + wk ≥ vj′ + wk′ for all (j , k ) ∈ Ai}, which is again defined by intersection of closed hyperspaces. Next, we show hungriness. If B(~v, ~w) 6= ∅, then at least one of these pairs, with longest slices in both cakes, is in Li, and both slices in this pair are nonempty. If B(~v, ~w) = ∅, then in cake 1 all slices are shorter than 1/(n − 1). Since their total length is 1, all slices in cake 1 are nonempty. For every max-sum pair (j, k) ∈ Ai, wk equals either w1 or wn. If both these lengths are positive then we are done. If one of them is zero, then the other must be at least 1/(n − 1), so the max-sum is at least than 1/(n − 1). Since all slices of cake 1 are shorter than 1/(n − 1), this max-sum can be attained only with a nonempty slice of cake 2. Finally, we show that in every partition (~v, ~w), at most n − 1 agents can be allocated an acceptable pair. Case 1: every agent i gets a pair (ji, ki) ∈ Ai. Since the largest matching in {(i, ji, ki) | i ∈ [2n − 2]}⊂ HD is of size n − 1, at most n − 1 agents are satisfied. 1 Case 2: at least one agent i gets a pair (ji, ki) ∈ B(~v, ~w) \ Ai. So vji ≥ n−1 and vji = maxj vj and wki = max(w1, wn). Cake 1 must have both large slices 1 1 (with length at least n−1 ) and small slices (with length less than n−1 ). We define 1 a maximal small-slice sequence as a sequence j1,...,j2 such that vj < n−1 for all 1 1 j1 ≤ j ≤ j2 while vj1−1 ≥ n−1 and vj2+1 ≥ n−1 . Subcase 2.1: wn = w1. Then, in every maximal small-slice sequence j1,...,j2, the only agents who are willing to accept a pair with a slice from this sequence are agents j1,...,j2 − 1: agent j1 − 1 won’t accept slice j1 since vj1 + wn < vj1 −1 + w1, and agent j2 won’t accept slice j2 since vj2 + w1 < vj2+1 + wn. Therefore, at least one of the slices in this sequence remains unallocated. FRACTIONALLY BALANCED HYPERGRAPHS 21 Subcase 2.2: wn 6= w1; suppose without loss of generality that w1 > wn. At least one agent i must get a pair (ji, ki) with n ≤ ki ≤ 2n − 2. Among all those agents, select one for which vji is smallest. Note that the pair (ji, ki) cannot come from B(~v, ~w) since wki ≤ wn so it is not maximum in cake 2. Therefore, the pair (ji, ki) must come from Ai, so we must have ji = i + 1 and vi+1 + wn ≥ vi + w1. This implies vi+1 > vi. The minimality of vji implies that slice i = ji − 1 is not assigned to agent i − 1 through Ai−1. It is also of course not assigned to agent i through Ai. It is also not assigned through B, since vi is not maximum in cake 1. So slice i remains unallocated. In all cases, at most n − 1 slices of cake 1 are allocated. 7. Rainbow matchings in families of d-intervals KKM-type theorems have been applied to prove results on matchings in d- interval families, see, e.g., [2, 9]. But it seems that the fact that there is a simple reduction of d-interval matching problems to multiple cake division problems hasn’t been explicitly stated. The purpose of this section is to note this reduction. It im- plies that results on multiple cake-division, in particular those proved above, yield lower bounds on matching numbers in d-interval hypergraphs. Given d disjoint copies C1,...,Cd of the unit interval [0, 1], a d-interval is the union of d disjoint open intervals, one on each Ct (the openness is assumed just for simplifying some arguments). Let F be a finite family of d-intervals. We think of F as a hypergraph whose vertex set is the uncountable set C1 ∪···∪ Cd and whose edges are the d-intervals. So a matching in F is a subset of F consisting of pairwise disjoint d-intervals, and a cover in F is a set of points in the vertex set intersecting all d-intervals in F. A well-known theorem of Gallai asserts that when d = 1, the matching number and the covering number in F are the same. For d ≥ 2, Tardos [24] and Kaiser [12] proved the following: Theorem 7.1 ([24, 12]). For d ≥ 2, any family of d-intervals with matching number m can be covered by d(d − 1)m points, (d − 1)m on each Ct. A rainbow version of this theorem was proved in [9]: Theorem 7.2. Let d ≥ 2. Let Fi, i ∈ [d(n − 1) + 1], are n families of d-intervals n and write F = i=1 Fi. If for all i ∈ [n], Fi cannot be covered by any choice of (n − 1)d points, (n − 1) on each Ct, then there exists a rainbow matching M in F S n (i.e., M is a matching in F and |M ∩ Fi|≤ 1) of size |M| ≥ d−1 . ad Our results on the relation −−→ imply further extensions of Theorem 7.2, an- swering questions of the form: Let F1,..., Fn be families of d-intervals, such that for every i ∈ [n], Fi cannot be covered by at − 1 points on each Ct. What is the largest rainbow matching we are guaranteed to find? im Notation 7.3. We write (n; a1,a2,...,ad) −−→ m if any n families of d-intervals F1,..., Fn, such that for every i ∈ [n] Fi cannot be covered by a choice of at − 1 points on each Ct, has a rainbow matching of size m. im n In this notation, Theorem 7.2 is (d(n − 1) + 1; n ··· ,n) −−→ d−1 . d times ad im Theorem 7.4. (n; a1,a2,...,ad) −−−→ m implies |(n{z; a1,a} 2,...,ad) −−→ m. 22 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB Proof. Given a collection of families Fi of d-intervals, we construct a d-cake-cutting instance in which each agent i accepts a d-tuple of pieces if and only if it contains a d-interval of Fi. Formally, for every (a1,...,ad)-partition P of the cakes, let C(P ) be the set of its cut-points — a set containing at − 1 points in each cake t. By assumption, for every i ∈ [n], Fi is not covered by C(P ). This means that Fi contains at least one d-interval Ji(P ) that is not cut by the partition. Every such Ji(P ) is entirely contained in some d-tuple of nonempty intervals from P , 1 d i say (Ij1 ,...,Ijd ). We let L (P ) consist of the corresponding vectors (j1,...,jd) of indices. The closedness of the admissibility sets follows from the fact that the d- intervals are open. An admissible division in the cake-cutting instance corresponds to a rainbow matching in the collection of d-interval families. As a sample application, here is a proof of the case d = 2 of Theorem 7.1. This special case is: in a family F of 2-intervals, if there is no cover containing k points in each of Ct, then there is a matching of size k + 1. ad By Theorem 5.1 there exists n such that (n; k +1, k + 1) −−→ k +1. (In fact, by Corollary 4.7 (1) n =2k + 1 suffices, but we prefer using Theorem 5.1 because it is a general tool for proving non-rainbow results from rainbow results.) Take n families, all identical to F . By Theorem 7.4, there exists a rainbow matching of size k + 1 of the union of these n families, which is a matching of size k + 1 in F . Remark 7.5. Possibly the reduction is only one way. It may well be that the bounds obtained this way are not optimal, namely that d-intervals behave better than division of d cakes. 8. Monotonicity The following proposition shows that results on admissible division of cakes are ad ad monotone in the sense that (a0; a1,...,ad) −−→ m implies (b0; b1,...,bd) −−→ m whenever bi ≥ ai for all i. ad Proposition 8.1. If (a0; a1,...,ad) −−−→ m and ∀i ∈ {0,...,d} : bi ≥ ai then ad (b0; b1,...,bd) −−−→ m. Proof. If b0 >a0, then one can just ignore b0 − a0 arbitrary agents and allocate to the remaining a0 agents. Due to symmetry and induction, it is sufficient to prove ad that (a0; a1,...,ad + 1) −−→ m. Let {pi : i ∈ [a0]} be a set of agents, each with a preference function defined on all (a1,...,ad + 1)-partitions of d cakes. Define a set {qi : i ∈ [a0]} of agents with preference functions on all (a1,...,ad)-partitions as follows. 1 d−1 d For each (a1,...,ad)-partition Q = (~x ,...,~x , ~x ), define an (a1,...,ad +1)- 1 d−1 d d d partition P (Q) = (~x ,...,~x , ~y ), where ~y equals ~x in its first ad coordinates, d and the (ad + 1)-st coordinate of ~y is 0. For each i, the preferences of agent qi in Q are determined by the preferences of agent pi in P : agent qi prefers in Q the k-tuple of slices with indices (j1,...,jd) iff jd ≤ ad and agent pi prefers in P the d-tuple of slices with indices (j1,...,jd). By the hungry agents assumption, and since slice ad + 1 is empty in P , agent pi must prefer in P at least one d-tuple with jd ≤ ad. The nonemptiness and continuity conditions for the preference lists of the qi’s follow from those of the pi’s. So by assumption, the {qi} have a satisfying assignment for m of the agents, which gives ′ rise to a corresponding assignment for m of the {pi} s. FRACTIONALLY BALANCED HYPERGRAPHS 23 bm We suspect that the corresponding monotonicity property for the“−−−→” relation is false. Question 8.2. Is it true that bm(6, 6, 5) = 4? This will refute monotonicity of bm, since bm(6, 6, 6) = 3 by Theorem 4.3. Acknowledgments We are grateful to Fr´ed´eric Meunier for helpful discussions. We also thank users Trebor, Tortar and Piquito from math stackexchange1 for their technical help. References [1] Michal Adamaszek and Jonathan Ariel Barmak. On a lower bound for the connectivity of the independence complex of a graph. Discrete Math., 311(21): 2566–2569, 2011. [2] Kaiser Tomas Aharoni, Ron and Shira Zerbib. Fractional covers and matchings in families of weighted d-intervals. Combinatorica, 37:555–572, 2017. [3] Ron Aharoni and Eli Berger. The intersection of a matroid and a simplicial complex. Transactions of the American Mathematical Society, 358(11):4895– 4917, 2006. [4] Ron Aharoni and Penny Haxell. Hall’s theorem for hypergraphs. J. Graph Theory, 35(2):83–88, 2000. [5] Ron Aharoni, Eli Berger, and Ran Ziv. Independent systems of representatives in weighted graphs. Combinatorica, 27(3):253–267, 2007. [6] Noga Alon and Kenneth A. Berman. Regular hypergraphs, gordan’s lemma, steinitz’ lemma and invariant theory. J. Combin. Theory, Ser. A, 43(11):91–97, 1986. [7] John Cloutier, Kathryn L. Nyman, and Francis E. Su. Mathematical Social Sciences, 59(1):26–37, January 2010. [8] Arthur A. Drisko. Transversals in row-latin rectangles. J. Combin. Theory, Ser. A, 84(11):181–195, 1998. [9] Florian Frick and Shira Zerbib. Colorful coverings of polytopes and piercing numbers of colorful d-intervals. Combinatorica, 39(3):627–637, 2019. [10] Zolt´an F¨uredi. Maximum degree and fractional matchings in uniform hyper- graphs. Combinatorica, 1(2):155–162, 1981. [11] David Gale. Equilibrium in a discrete exchange economy with money. Internat. J. Game Theory, 13(1):61–64, 1984. [12] Tomas Kaiser. Transversals of d-intervals. Discrete Comput. Geom., 18(2): 195–203, 1997. [13] D´enes K˝onig. Graphok ´es alkalmaz´asuk a determin´ansok ´es a halmazok elm´elet´ere. Mathematikai ´es Term´eszettudom´anyi Ertesito, 34:104–119, 1916. [14] Bronislaw Knaster, Kazimierz Kuratowski, and Stefan Mazurkiewicz. Ein be- weis des fixpunktsatzes f¨r n-dimensionale simplexe (in german). Fundamenta Mathematicae, 14:132–137, 1929. [15] Hidetoshi Komiya. A simple proof of K-K-M-S theorem. Econom. Theory, 4 (3):463–466, 1994. 1https://math.stackexchange.com/q/3811677/29780 24 AHARONI, BERGER, BRIGGS, SEGAL-HALEVI, AND ZERBIB [16] Nicolas Lebert, Fr´ed´eric Meunier, and Quentin Carbonneaux. Envy-free two- player m-cake and three-player two-cake divisions. Operations Research Let- ters, 41(6):607–610, November 2013. [17] Roy Meshulam. The clique complex and hypergraph matching. Combinatorica, 21(1):89–94, 2001. [18] Roy Meshulam. Domination numbers and homology. J. Combin. Theory Ser. A, 102(2):321–330, 2003. [19] Fr´ed´eric Meunier and Francis Edward Su. Multilabeled versions of sperner’s and fan’s lemmas and applications. SIAM Journal on Applied Algebra and Geometry, 3(3):391–411, 2019. [20] Kathryn Nyman, Francis Edward Su, and Shira Zerbib. Fair division with multiple pieces. Discrete Applied Mathematics, 283:115–122, 2020. [21] Erel Segal-Halevi. Fairly Dividing a Cake after Some Parts Were Burnt in the Oven. In Proceedings of AAMAS’18, April 2018. [22] Walter Stromquist. How to cut a cake fairly. American Mathematical Monthly, 87(8):640–644, 1980. [23] Francis Edward Su. Rental harmony: Sperner’s lemma in fair division. Amer- ican Mathematical Monthly, 106(10):930–942, 1999. [24] G´abor Tardos. Transversals of 2-intervals, a topological approach. Combina- torica, 15(1):123–134, 1995. [25] Douglas R. Woodall. Dividing a cake fairly. Journal of Mathematical Analysis and Applications, 78(1):233–247, 1980. Appendix A. Notation summary n Num. of agents i Index of an agent; i ∈ [n]. d Num. of cakes / intervals t Index of a cake / interval; t ∈ [d]. at Num. of pieces of cake t jt Piece-index in cake t; jt ∈ [at]. d d P = t=1 ∆at−1; set of d-cake partitions. J = t=1[at]; set of piece-index vectors t t P partition of cake t; P ∈ ∆at−1 P partition of d cakes; P ∈P Q t Q Ct cake t / interval t (t ∈ [d]) Ij (P ) Interval j in partition P of cake t bm Matching in frac. balanced hyp. im Matching in interval hyp. ad Admissible cake-division The following implications hold: bm ad im An arrow 7→ means that we know that the opposite implication does not hold. When an arrow → is drawn, it is open whether the opposite implication holds.