Choice Functions Is Given by Two Theorems

Home , Pigeonhole principle

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The sets in are thought to “color” the elements, hence the name “rainbow”. Hall’s S marriage theorem (anticipated by theorems of Frobenius and K¨onig) is:

Theorem 1.3. Let be a family of sets. If ′ ′ for every ′ , then there S | S S | ≥ |S | S ⊆ S exists a full injective choice function.

The conjecture to which we allured is usually stated in the terminology of Latin squares. An n n matrix L is called a Latin square if its entries belong to [n] := 1,...,n , and × { } every row and every column consists of distinct elements. A (partial) transversal in L is a set of distinct entries, lying in diﬀerent rows and diﬀerent columns. It is called full if its size is n.

Conjecture 1.4. [21, 49] Every n n Latin square has a transversal of size n 1. If n is × − odd, then there exists a full transversal.

The best result to date is by Keevash-Pokrovsky-Sudakov-Yepremyan:

Theorem 1.5. [39] Every n n Latin square has a transversal of size n O(log n/ log log n). × −

The paper is not presumed to be comprehensive in any way, but rather to touch on some main themes. It also contains a few new results: on “scrambled” versions (to be deﬁned in Section 8), on cooperative versions (Section 11) and on weighted versions (Section 10).

2. A general theme - big in one sense, small in another

Hall’s marriage theorem concerns a well-known theme: the existence of an object satisfying two opposing requirements. It should be large in one sense, and small in another. For example, in the marriage theorem the choice function should be large on the side of the sets - represent all of them, and small on the elements side - being injective. A hypergraph H is called a simplicial complex, or just a complex if it is closed down, namely e H, f e imply f H. A simplicial complex is a matroid if all its maximal ∈ ⊆ ∈

2 edges are also of maximal size, and this is true also for every induced sub-complex. An edge of the matroid is said to be independent, a set containing a maximal edge is said to be spanning, and a set that is both independent and spanning is a base. Given a complex C we think of its elements as “small”, and a spanning set in a matroid is considered “large”. Of particular interest from a combinatorial point of view are partition matroids, deﬁned by a partition of the ground set, where a set belongs to the matroid if it meets every part in at most one vertex. A basis of a partition matroid is just a full rainbow set of the parts. In a more general setting a matroid is given on V := , and to the injectivity M S F requirement a requirement is added that the rainbow set R satisﬁes R . In Hall’s ∈ M theorem is the partition matroid whose parts are the v-stars, v V . Hall’s theorem M ∈ then generalizes to the following [47]:

Theorem 2.1 (Rado). There exists a full rainbow set R belonging to if and only if M rank ( ) I for every I [m]. M FI ≥| | ⊆ Edmonds [30] oﬀered a further generalization. Consider the bipartite graph whose one side is and the other is , where S is connected to x if x S. The two S S S ∈ S ∈ S S ∈ matroids are the two partition matroids on the edge set E of the graph, whose parts are the stars in the two respective sides. In this setting, a set of edges contains a full choice function if it is spanning in the ﬁrst matroid, and it is injective if it belongs to the second matroid. In a further generalization, one of the matroids is replaced by a general simplicial complex.

Deﬁnition 2.2. Let , be a matroid and a complex, respectively, on the same ground M C set. An ( , )-marriage is a set spanning in and belonging to . An( , )-rainbow M C M C M C set is a set belonging . M \ C The “if” direction of Theorem 2.1 still holds in this setting, with “rank” replaced by “coonectivity”, to be deﬁned below. We shall be particularly interested in rainbow matchings. Given a hypergraph on a set V and a partition of its edge set, a partial rainbow matching is a partial choice function on the parts, whose image is a matching (=set of disjoint edges). A more general notion is that of rainbow independent sets in a graph. Matchings are the special case, of independent sets in a line graph.

3 Given a class of hypergraphs, we write (a, b) C c if every family of a matchings of C → size b in a hypergraph belonging to has a rainbow matching of size c. Let be the class C G of all graphs, and the class of bipartite graphs. B Conjecture 1.4 has a generalization, by Berger and the ﬁrst author, that can be formulated in this terminology. We shall refer to it as the “A-B conjecture”:

Conjecture 2.3. [1] (n, n) G n 1. → −

In words: n matchings of size n in any graph have a rainbow matching of size n 1. − In [1] this was formulated only for bipartite graphs, but we know no counterexample also for general graphs. In the bipartite case, doubling the number of matchings in the condition yields a well- known theorem of Drisko:

Theorem 2.4. [29, 1] (2n 1, n) B n. − →

In words: 2n 1 matchings of size n in a bipartite graph have a rainbow matching of − size n. A very nice recent result, using algebra for its proof, is:

Theorem 2.5. [27] Let be the class of r-uniform hypergraphs. Cr t. Cr →

Other conjectures in the spirit of Conjecture 2.3 are:

Conjecture 2.6.

(1) (2n, n) G n. If n is odd then (2n 1, n) G n. → − → (2) (n, n + 1) B n. → (3) (n, n + 2) G n. →

An example showing that (n.n + 1) G n is given by three matchings M , i 3 6→ i ≤ on a set a, b, c, d, a′, b′,c′,d′ , where M = ab, cd, a′b′,c′d′ , M = ac, bd, a′c′, b′d′ and { } 1 { } 2 { } M = ad, bc, a′d′, b′c′ . However, we do not know of any such example for larger n. 3 { } Pokrovskiy [45] proved that (n, n + o(n)) B n and Rao, Ramadurai, Wanless and → Wormald [32] gave better asymptotics in the case of simple graphs. Keevash and Yepremyan [40] proved that n matchings of size n + o(n), repeating each edge at most o(n) times, have a rainbow matching of size n const. It is not hard to show that (n, n) B n o(n), while − → − the same statement for general graphs is still open, see Section 4 below.

4 Interestingly, part (2) of Conjecture 2.6 implies Theorem 2.4 (of course, the other direction would be more desirable). Given 2n 1 matchings of size n in a bipartite graph, − double each of them by joining it to its copy on a disjoint vertex set V ′. We now have at hand 2n 1 matchings of size 2n, so assuming (2n 1, 2n) B 2n 1, we get a rainbow − − → − matching of size 2n 1, of which by the pigeonhole principle n edges must be in the same − copy - V or V ′. Similarly, Conjecture 2.3 implies part (1) of the conjecture. For bipartite graphs, (2) implies (1). If true, (1) is a strengthening of Conjecture 1.4. To see this, note that for every i [n] the set of entries that are equal to i forms a matching ∈ between the rows and the columns. For a non-decreasing sequence σ = (a ,...,a ) of natural numbers write σ n if for 1 k → every set of families F of matchings in a bipartite graph, where F = a , there exists a i | i| i rainbow matching of size n. We say then that σ is n-coercive. So, Conjecture 2.3 is that (n +1,...,n + 1) n ((n + 1)-fold repetition). We do not even know a counterexample → to (n, . . . n, , n +1,...,n + 1) n (n, n + 1 respectively repeated n/2 , n/2 times). → ⌈ ⌉ ⌊ ⌋ In [5] the following strengthening of Theorem 2.4 was proved:

Theorem 2.7. (1, 2,...n,n,...,n) n (n repeated n times). →

The only known proof is topological. This sequence is minimal n-coercing, in the following sense: reducing any term makes the sequence non-n-coercing. Another example of a minimal n-coercing sequence is (1, 3,..., 2n 3, 2n 1). − −

Conjecture 2.8.

(1) If σ is minimal n-coercing then its length is at most 2n 1. − (2) An n-coercing sequence contains an n- coercing subsequence of length at most 2n 1. − (3) If σ is not n-coercing then there is a set of matchings witnessing this, all contained in a disjoint union of cycles. (4) If (a ....,a ) n then k a n2. 1 k → Pi=1 i ≥

The sequence (2, 4, 4) is not 3-coercing, and the only example witnessing this is a disjoint union of two cycles of length 4, so part (3) of the conjecture cannot be strenthened to “one cycle”. Conjecture 2.3 is easily seen to be true in cycles.

5 3. Two dual, topologically-formulated, theorems

The rank of a complex is the largest size of an edge. A complex of rank k has a unique (up to homeomorphism) geometric realization in R2k−1 - think of the realization of a graph (k = 2) in R3. A parameter connecting topology and combinatorics is (homotopic) connectivity, which is the minimal dimension of a hole in the geometric realization. Its homological version is the minimal index of a non-zero homology group, plus 1. The two are not identical - homological connectivity is at least as large as homotopic connectivity. The homotopic connectivity is denoted by η( )(η = if there are no holes). Rainbow problems C ∞ and marriage problems give rise to two mirror requirements on the holes. In the ﬁrst, it is beneﬁcial (namely contributing to the existence of rainbow sets) not to have small dimensional holes. In the second, it is useful not to have large dimensional holes. We denote by λ( ) the largest dimension of a hole (0, if there is no hole), and by λ¯( ) the C C maximum of λ( [S]) over all subsets S of V ( ). It is called the “Lerayness” of . C C C Example 3.1. Let Sn be (a triangulation of) the n-dimensional sphere, and Bn (a triangulation of) the n-dimensional ball. Then η(Sn)= λ(Sn)= n +1, η(Bn)= . ∞ Remark 3.2. It can be shown that η is the minimal dimension of an empty sphere. By contrast, λ is not necessarily the largest dimension of an empty sphere - the “holes” can be more general. For example, for a torus λ = 3, while η = 2, since there is an empty S1.

The two parameters are connected via an operation called “Alexander duality”. The Alexander dual D( ) of a complex is Ac A . C C { | 6∈ C} Theorem 3.3 (e.g. [20]). λ¯( )= V ( ) η(D(C)) 1. C | C | − − The connection between these notions and choice functions is given by two theorems. One is “Topological Hall” (implicit in [13], ﬁrst stated explicitly, quoting the ﬁrst author, in [44]), whose matroidal form is:

Theorem 3.4. [2] If η( [S]) rank( .S) for every S V then there exists a ( , )- C ≥ M ⊆ M S marriage.

Here .S is the contraction of to S. “Topological hall” is the case in which is M M M a partition matroid. (The disjointness of the parts in the partition matroid is attained by duplicating vertices, if necessary.)

6 The fundamental topological theorem regarding the second type of choice functions is that of Kalai-Meshulam, a special case of which is:

Theorem 3.5 (Kalai–Meshulam [38]). If λ¯( ) d then every d +1 sets in c := 2V C ≤ C \ C have a rainbow set belonging to c. C

As in Theorem 2.4 and Conjecture 2.3, the format is that of “many sets not in have C a rainbow set not in ”. Theorems 3.4 and 3.5 are derivable from one another, using C Theorem 3.3 and noting that A if and only if Ac is spanning in ∗ and belongs ∈M\C M to D(C). ( ∗, the dual of , consists of those sets whose complements span ). M M M A classical theorem of this type is the Bárány-Lovász colorful version of Caratheodory’s theorem. For a set V = ~v , i I of vectors in Rd let { i ∈ }

cone(V )= X αi~vi, αi 0 for all i I { ≥ ∈ } i∈I and

conv(V )= X αi~vi, αi 0 for all i I, X αi =1 { ≥ ∈ } i∈I i

Theorem 3.6. [B´ar´any, [17]]

(1) If S ,...,S are sets of vectors in Rd satisfying ~v cone(S ) for all i d then 1 d ∈ i ≤ there exists a rainbow set S such that ~v cone(S). ∈ (2) If S ,...,S are sets of vectors in Rd satisfying ~v conv(S ) for all i d +1 1 d+1 ∈ i ≤ then there exists a rainbow set S such that ~v conv(S). ∈

The complex consists here of sets not containing ~v in their convex hull (or cone). The C case of Theorem 2.4 in which the matchings reside in Kn,n can be derived from Theorem 3.6 - we omit the proof.

4. General graphs

Recall Conjecture 2.6 (1): (2n, n) G n. The fractional version of this conjecture was → proved in [14]. It says that any 2n matchings of size n, in a general graph, have a rainbow set of edges possessing a fractional matching of size n. The best result so far is (3n 3, n) n − → [9]. Holmsen and Lee [36] gave a topological proof to the weaker result (3n 2, n) n, − → that is stronger in another sense - it yields a cooperative version:

7 Theorem 4.1. Any collection of 3n 2 non-empty sets of edges in any graph, satisfying − the condition that every pair of them contains in its union a matching of size n, has a rainbow matching of size n,

We shall return to the theme of cooperation in Section 11. Recently, Chakraborti and Loh [25] proved an asymptotic version of the conjecture, when the matchings are disjoint. Recall also Conjecture 2.3: (n, n) G n 1. Woolbright [50] proved (n, n) B n √n. → − → − But in general graphs the situation is gloomier: we do not even know how to show (n, n) G → n o(n). The best result so far is in [11], the existence of a rainbow matching of size at least − 2n 1. The result, with √2n replacing o(n), would follow from the following conjecture: 3 −

Conjecture 4.2. Let F be a matching in a graph, and let be families of disjoint aug- Fi menting F -alternating paths. If 2 F then there exists a set K of edges, each P |Fi| ≥ | | taken from a path in a diﬀerent , that contains an augmenting F -alternating path. Fi

A proof of this fact when the s are each a singleton path is at the core of the proof in Fi [4]. The non-alternating version of this conjecture was proved in [11]:

Theorem 4.3. Let F be a set of vertices in a graph, and let be families of disjoint Fi paths, starting and ending outside F . If 2 F then there exists a set K of edges, P |Fi| ≥ | | each taken from a path in a diﬀerent , that contains a path starting and ending outside Fi S F .

The conjecture would follow from (n, n + o(n)) G n, but presently this is known only → for sets of edge-disjoint matchings.

5. Covering the vertex set by rainbow sets

The next step goes one level higher: asking for many disjoint full rainbow sets. The best known conjecture of this sort is Rota’s conjecture, that says that given n independent sets of size n in a matroid , there are n disjoint -independent rainbow sets. A recent M M startling development on the conjecture was its asymptotic proof by Pokrovskiy [46] - there are n partial rainbow independent sets covering n2 o(n) of the elements. − For a hypergraph H, denote by ρ(H) the minimal number of edges covering V (H). A conceivable strengthening of Rota’s conjecture is that the sets need not be in - it may M

8 suﬃce that they are in only after scrambling. Namely - does it suﬃce to assume that M ρ( ) n? In [15] counterexamples were given for n odd, but the following may be true: M ≤

Conjecture 5.1.

(1) If ρ( ) = n and V ( ) is partitioned into n sets of size n, then V ( ) can be M M M partitioned into n +1 rainbow -sets. M (2) If n is even then n rainbow -sets suﬃce. M

Admittedly, the n-even part of the conjecture is rather daring, and is safer to be conjectured just for linear matroids. But it matches a diﬀerence in the original conjecture: in the linear case, Rota’s conjecture is known for inﬁnitely many even n’s, while for n odd it is known only for n = 3 (a result of R. Huang). Part (1) of the conjecture is a special case of:

Conjecture 5.2. If , are two matroids on the same ground set, then ρ( ) M N M∩N ≤ max(ρ( ), ρ( ))+1. M N

Conjecture 5.1(1) is the case in which one of the two matroids is a partition matroid. The conjecture is known even without the “+1”, when and are truncated partition M N matroids, namely the intersection of a partition matroid with a uniform matroid (the latter being the collection of sets of size k for some k) - this is a re-formulation of K¨onig’s ≤ edge-coloring theorem. In particular, this is true when rank( ), rank( ) 2 - it is easy M N ≤ to show that a rank 2 matroid is of this type. It is not hard to show that if , are matroids, then max(ρ( ), ρ( )) = ρ∗( ), M N M N M∩N and therefore Conjecture 5.2 can be re-formulated as:

Conjecture 5.3. ρ( ) ρ∗( )+1 M∩N ≤ M∩N

This is a close relative of the famous Goldberg-Seymour conjecture, stating the same inequality where the intersection of two matroids is replaced by the matching complex of a multigraph. The Goldberg-Seymour conjecture has recently been claimed to be solved [26], and the methods used there (mainly relying on alternating paths) may be relevant also to this conjecture. In fact, there is a common generalization to this conjecture and the Goldberg-Seymour conjecture. Its objects are 2-polymatroids. A hypergraph on a vertex set V is called a P

9 2-polymatroid if there exists a matroid is on V 1, 2 , such that = A A 1, 2 M ×{ } P { | ×{ } ∈ . In [6] the following was proposed: M}

Conjecture 5.4. If every circuit (minimal non- set) meets every pair (v, 1), (v, 2) in M { } at most one element, then ρ( ) ρ∗( )+1. P ≤ P

In [6] this is proved when ρ∗( ) 2. P ≤ In [2] the following was proved:

Theorem 5.5. ρ( ) 2 max(ρ(M), ρ(N)). M∩N ≤

Can an analogous packing result (rather than covering) be proved? It is easy to prove that there exists a set in of size |V | . Can we prove the existence of M∩N ⌈ max(ρ(M),ρ(N)) ⌉ “many” disjoint such sets? A beautiful strengthening of Theorem 5.5 was suggested in [[19], Conjecture 1.10]:

Conjecture 5.6. Every matroid contains a partition matroid with ρ( ) 2ρ( ). M P P ≤ M

Theorem 5.5 will follow from this conjecture by K¨onig’s edge-coloring theorem. Scrambling takes us to a happy hunting ground, with an abundance of problems. Can we prove the existence of two disjoint ( , )-marriages in the scrambled setting of Rota? M P n In the original setting of Rota’s conjecture it is known [22] that there exist 2 disjoint such marriages. Matchings are independent sets in line graphs, and our problems can be generalized, to independent rainbow sets in general graphs. A well-known conjecture on such sets (see, e.g., [35]) is:

Conjecture 5.7. Let G a graph with maximal degree d, and let be a partition of V (G) S into sets of size 2d. Then V (G) can be covered by 2d independent rainbow sets.

In [10] the following generic question was studied: given a class of graphs, how many independent sets of size n do you need to procure a rainbow independent set of size m? The following conjecture was posed there, with some partial results:

Conjecture 5.8. k+1 (m 1)+1 independent sets of size n in a graph with maximal n−m+2 − degree k have a rainbow independent set of size m.

10 Conjecture 5.8 is open even for k = 2, in which case independent sets can be viewed as matchings - taking us back to rainbow matchings. The special case k = 2, m = n says that n independent sets of size n in the disjoint union of cycles and paths have a rainbow independent set of size n 1. Switching from m = n 1 to m = n causes the expression − − k+1 to jump from 3 =1to 3 = 2. This is one point of view from which to explain ⌈ n−m+2 ⌉ ⌈ 3 ⌉ ⌈ 2 ⌉ the jump from n to 2n 1 matchings of size n, when the desired size of the rainbow − matching goes from n 1 to n. − In [10] examples were given to show that, if true, the conjecture is sharp. Also “half” of the conjecture is proved - if m n then k(m 1) + 1 independent sets of size n in a graph ≤ − of maximal degree k have a rainbow independent set of size m.

6. Rainbow cycles and rainbow spanning sets

We next turn to problems of algebraic nature. The following fact can be proved using Theorem 3.5, but it also has an easy direct proof:

Observation 6.1. If rank( ) n then any collection of n +1 sets not belonging to has C ≤ C a rainbow set not belonging to . C

A bit less trivial:

Theorem 6.2. Any family =(A ,...,A ) of edge sets of odd cycles on a set of size n A 1 n has a rainbow odd cycle.

This is a special case of a more general result:

Theorem 6.3. Let be a matroid of rank n and let v V ( ). Any collection A ,...,A M ∈ M 1 n of sets spanning v has a rainbow set spanning v.

Theorem 6.2 follows from Theorem 6.3 and the following lemma:

Lemma 6.4. G is bipartite if and only if

(~0, 1) span( (χ , 1) e E(G) ) 6∈ { e | ∈ } Fn Here the vectors are taken in 2 . Note that the condition is just that of aﬃne spanning. To prove Theorem 6.2, take the matroid in Theorem 6.3 to be the linear matroid M over F2 and replace every edge e by the vector (χe, 1).

11 For n odd, a Hamilton cycle on n vertices, taken n 1 times, shows that Theorem 6.2 is − sharp, namely n 1 odd cycles do not suﬃce. It was shown in [8] that the only examples − showing sharpness are cacti-like graphs, generalizing the above example. One of the best-known conjectures in graph theory is the Caccetta - H¨aggkvist conjecture:

n Conjecture 6.5. [23] In a digraph on n vertices and minimum out-degree at least r there is a directed cycle of length r or less.

In [12] this was given a rainbow generalization.

Conjecture 6.6. In an undirected graph on n vertices, any collection of n disjoint sets of n edges, each of size at least r , has a rainbow cycle of length r or less.

In [28] the conjecture is proved for r = 2, while the original C-H conjecture is known up to r = 5 - so there is a lot of room for progress.

7. Rainbow paths in networks and their relation to rainbow matchings

A network is a triple (D,S,T ), where D is a digraph on a vertex set V , and S, T are disjoint sets, of “sources” and “targets”, respectively It is assumed that no edge goes into s, and no edge goes out of t. In [16] the following result was proved, as a step towards proving Theorem 2.4:

Theorem 7.1. If V (D) (S T ) = n then any family of n +1 S T -paths has a | \ ∪ | F − rainbow S T path. − The two theorems have a common generalization. Given a set of edges let νP ( ) be F F the maximal size of a family of S T disjoint paths with edges from . − F Theorem 7.2. Let be a network with V ◦( ) = q, and let p be an integer. Let N | N | =(F ,...,F ) be a family of sets of edges, satisfying νP (F ) p for all i 2p 1+q. F 1 2p−1+q i ≥ ≤ − Then there exists an -rainbow set R with νP (R) p. F ≥ The case q = 0 is Theorem 2.4, and the case p = 1 is Theorem 7.1. The proof requires some deﬁnitions. Given a network , deﬁne a bipartite graph B( ) whose vertex set is v′ v S N N { | ∈ ∪ V ◦ v′′ v T V ◦ , and whose edge set is u′v′′ uv E( ) W , where }∪{ | ∈ ∪ } { | ∈ N } ∪ W = x′x′′ x V ◦ . { | ∈ }

12 Here v′ is a “sending” copy of v, and v′′ is an “absorbing” copy of v. For a matching M in B( ) let ψ(M)= M W . N \ A linearish arborescence is a digraph in which every vertex has in-degree and out-degree at most one, meaning that it is a collection of directed paths and cycles. Given a linearish arborescence L in , let φ(L) be the matching x′y′′ xy E(L) x′x′′ x V (L) in N { | ∈ }∪{ | 6∈ } B( ). N For a linearish arborescence L let LCIRC be the the set of cycles in L, LS the set of paths in L starting at S, L the set of paths in L ending at T , L = L L . By deﬁnition, T ST S ∩ T νP (L) = L . | | | ST | Claim 7.3. L = φ(L) V ◦( ) + L (L L ) . | ST | | |−| N | | \ S ∪ T | To see this, note: (1) If C L then V (C) = E(C) , ∈ CIRC | | | | (2) if P L then E(P ) = V ◦(P ) +1, so P contributes to φ(L) 1 more than to ∈ ST | | | | V ◦, (3) paths in (L L ) L contribute the same number to φ(L) and to V ◦, and S ∪ T \ ST (4) paths in L (L L ) contribute 1 less to φ(L) than to V ◦. \ S ∪ T Proof of Theorem 7.2. For every i 2p 1+q let M = φ(F ). By Claim 7.3 (taking L = F ) ≤ − i i i M = p+q. Consider the sequence of 2(p+q) 1 matchings W , W ,...W , M ,...,M , | i| − 1 2 q 1 2p−1+q where W = W for all j q. By Theorem 2.7 (letting n = p + q), these matchings have a j ≤ rainbow matching R of size p + q. Let L = ψ(R). By Claim 7.3 νP (R) F q = p, and ≥| i| − clearly the paths in R witnessing this are constructed from edges in Mi.

8. Scrambling

We next return to a notion from Section 5:

Deﬁnition 8.1. Given two families (that is, multisets) , of subsets of the same ground F S set, we say that is a scrambling of if = (as multisets), and that it is an S F S F S S n-scrambling if every set in has size at most n. S A straightforward application of Rado’s theorem (Theorem 2.1) yields:

Theorem 8.2. An n-scrambling of n independent sets of size n in a matroid have an independent full rainbow set.

13 In [3] the following was proved:

Theorem 8.3. If is a family of n2 n/2 matchings of size n in a bipartite graph and F − is an n-scrambling of , then has a rainbow matching of size n. S F S The proof uses Topological Hall. This may possibly be improved:

Conjecture 8.4. Let n 4. If is a family of n +1 matchings of size n in a bipartite ≥ F 2 graph and is an n-scrambling of , then has a rainbow matching of size n. S F S If true, this is best possible. In [3] the following was shown:

Theorem 8.5. For n 4, there exists an n-scrambling of n matchings of size n in a ≥ 2 bipartite graph, with no rainbow matching of size n.

We can prove the following scrambled version of Theorem 7.1:

Theorem 8.6. Suppose V (D) s, t = k, and is a family of > nk s t paths. If ′ | \{ }| F 2 − F is an n-scrambling of then ′ has a rainbow s t path. F F − The main tool in the proof will be the following class of objects.

Deﬁnition 8.7. An n-fold source tower in D is a sub-digraph S containing s, equipped

with an ordering of its vertices s = v0, v1,... such that the set of edges to vi from all − previous vj, denoted S (vi), is of size at least n for every i > 0. In particular, there are n edges from s to v . Similarly, T D containing t is an n-fold target tower if V (T ) has ≥ 1 ⊂ an ordering where every u T has a set T +(u) of at least n edges to all previous vertices. ∈ Proof. (Theorem 8.6): Let S, T be a maximal pair of vertex-disjoint n-fold source and target towers. Let W = V (D) (S T ). For each w W , denote by e(S,w), e(w, T ) the sets of edges \ ∪ ∈ in D from S to w and from w to T respectively. Suppose ﬁrst that there is some w W with e(S,w) + e(w, T ) > n. Note that both ∈ | | | | e(S,w) and e(w, T ) are nonempty, otherwise the other would have size at least n + 1 and w could be added to the corresponding n-fold tree, contradicting maximality. Consider the family of edge sets

:= S−(v ), ..., S−(v ), T +(u ),...,T +(u ), e(S,w), e(w, T ) . K { 1 a 1 b }

14 (here a and b are sizes of the source tower and target tower, respectively.) has the following two properties: K Any full choice function for contains an s t path, and • K − For any ′ , ′ n( ′ 1) + 1. • K ⊆K | S K | ≥ |K | − The ﬁrst condition arises since a choice function for S−(v ) is a directed tree spanning { i } V (S) and rooted at s, a choice function for T +(u ) is a tree spanning V (T ) directed { i } towards t, and a choice function for e(S,w), e(w, T is a path of length 2 from S to T . { } The second follows since all sets in have size at least n, except for e(S,w) and e(w, T ) K which have at least n + 1 combined (and crucially, both are at least 1). Call a family with these two properties a path enforcer. Note that any path enforcer K contains a rainbow s t path as needed: pigeonhole implies that every ′ contains − K ⊂ K edges of at least ′ diﬀerent colours in ′ (as it is an n-scrambling of ), and Hall’s |K | F F marriage theorem then implies there is a choice function for which is ′-rainbow, which K F in turn contains an s t path. − So instead, we may assume e(S,w) + e(w, T ) n for every w. Since there are > nk/2 | | | | ≤ paths in D leaving S (and entering T ), we can double-count as follows:

nk < X e(S,w) + X e(w, T ) | | | | w6∈S w6∈T

= X ( e(S,w) + e(w, T ) )+2 e(S, T ) | | | | | | w∈W n W +2 e(S, T ) ≤ | | | | nk +2 e(S, T ) , ≤ | | meaning there is at least one edge e from S to T . But now, S−(v ),...,S−(v ), T +(u ),...,T +(u ), e is a path enforcer, so again it { 1 a 1 b { }} contains an s t path which is ′-rainbow. − F

9. Adding a size condition

In [1] (Theorem 4.1) the following strengthening of Drisko’s theorem was proved:

Theorem 9.1. Let k n, and let M ,...,M be a set of 2k 1 matchings, each of ≤ 1 2k−1 − size n. Then there exists a matching of size n contained in S Mi, representing at least k matchings Mi.

15 Note that here, “representing” is injective, namely each representation is by a distinct edge. It may also be possible to strengthen Theorem 9.1 along the lines of Theorem 2.7.

Conjecture 9.2. Let k n, and let M ,...,M be a set of 2k 1 matchings, where ≤ 1 2k−1 − M = min(i, k 1) for i k 1 and M = n for k i 2k 1. Then there exists a | i| − ≤ − | i| ≤ ≤ − matching of size n contained in S Mi, representing at least k matchings Mi.

Proof of the case k =2. We have three matchings, M1, M2, M3, where M1 is a singleton edge e , and M = M = n. If M M contains two disjoint cycles, then taking the { } | 2| | 3| 2 ∪ 3 M2 edges from one of them and the M3 edges from the other yields the desired matching. If M M consists of a single cycle C, then e partitions C into two parts, and taking 2 ∪ 3 the M2 edges on one side, the M3 edges on the other, and adding e, results in the desired matching.

10. Weighted versions

Theorem 10.1 below is a weighted generalization of Theorem 7.1. Given an edge- weighting w of D and a subset E of its edges, let the weight of E, written w(E), be

Pe∈E w(e).

Theorem 10.1. If V (D) s, t = n then any family of n +1 s t-paths of weight | \{ }| F − k has a rainbow s t path of weight k. ≤ − ≤ As a special case, note that a collection of n +1 s t paths of length at most k have a − rainbow s t path of length at most k (as seen by giving a weight of 1 to every edge). − Throughout the proof, whenever a vertex v is in a tree T rooted at s, we will write T v for the directed path in T from s to v.

Proof. Construct a nested sequence of s-rooted trees

s = T T T T = T { } 0 ⊂ 1 ⊂ 2 ⊂···⊂ a as follows. For each i, if t T , the sequence terminates. Otherwise, let be the collection ∈ i Fi of all paths in not used in T (as t has not yet been reached, at most n of the n + 1 paths F i in are currently represented, so = ). Then let T be the tree obtained from T by F Fi 6 ∅ i+1 i adding the edge e starting at some vertex v V (T ) which minimizes the quantity i ∈ S Fi ∈ i w(Tiv)+ w(ei).

16 The sequence terminates at some tree T containing t. Since e for each i, every T i ∈Fi i is certainly -rainbow, so in particular T contains a rainbow s t path T t. F − Claim also, whenever P (i.e. P is a path currently unrepresented in T ) and u ∈ Fi i ∈ V (T ) V (P ), that i ∩ w(T u) w(Pu). i ≤

The theorem follows, as if the ﬁnal edge e = ea reaching t in T had out-vertex u and represented the path P , then ∈F w(T t)= w(T u)+ w(e) w(Pu)+ w(e)= w(P t) k. a−1 ≤ ≤ Prove the claim by induction on i. If u = s, then both paths are empty and their weights are 0. If u = s, then there is an edge e = vu T for some j < i which represented a diﬀerent 6 j ∈ i path Q . ∈F Moreover, P contains a path from s T to u T , so let x be the last vertex before u ∈ j 6∈ j on P which is also in Tj, and y be the vertex following x on P (so possibly y = u). Then w(T v)+ w(e ) w(T x)+ w(xy) by choice of e , and w(T x) w(P x) by induction (as P i i ≤ j i j ≤ was certainly not yet represented in Tj). Hence

w(T u)= w(T v)+ w(e ) w(T x)+ w(xy) w(P x)+ w(xy)= w(Py) w(Pu), i i i ≤ j ≤ ≤ where the last inequality follows from nonnegativity of w, as desired.

A corresponding weighted version of Theorem 2.4 may also be true:

Conjecture 10.2. Let G be a bipartite graph with an edge-weighting. Then 2n 1 match- − ings of size n and weight k in G have a rainbow matching of size n and weight k. ≤ ≤

Note that in this instance, the problem is invariant under affine transformations, so dropping the non-negativity assumption on the weighting does not make the problem any harder (The weighted version of Theorem 10.1, by contrast, has small counterexamples). Given that the original proof [29] of Theorem 2.4 was via Bárány’s Theorem (Theorem 3.6), one may also ask whether a weighted version also holds-in Rd-do d +1 ~v-convexing sets in of weight k have a rainbow ~v-convexing set of weight k? ≤ ≤

17 11. Cooperative versions

Hall’s theorem is “cooperative” in the sense that its assumption is not on individual sets, but on unions of sets. There are also cooperative versions of the second type of choice functions results - spanning rainbow sets. Here is a natural conjectured cooperative version of Theorem 2.4, which also generalizes Theorem 2.7:

Conjecture 11.1. Let = (F ,...,F ) be a system of sets of edges in a bipartite F 1 2k−1 graph. If ν( ) min( I ,k) for every I [2k 1] then there exists a rainbow matching FI ≥ | | ⊆ − of size k.

Theorem 2.4 is the special case where ν(F ) k for every i. The cooperative version in i ≥ which ν(F ) k whenever i = j, is Theorem 11.6 below. {i,j} ≥ 6 A fractional version of the conjecture was proved in [5]. A topological version is:

Conjecture 11.2. Let G be a bipartite graph, and let M(G) be the matching complex of G. Suppose that one side of the graph is of size 2n 1, and that every k vertices in that − side have at least min(k, n) neighbors in the other side. Then η(M(G)) n. ≥ Here is a cooperative conjecture on networks. Given a set F E( ) let R(F ) be the ⊆ N set of vertices x having an F -path from s to x.

Conjecture 11.3. Let n = V ◦(D) , and let =(F ,...,F ) be a family of sets of edges | | F 1 n+1 in the network. If, for every I [n + 1], either contains an s t-path or R( ) I ⊆ FI − | FI |≥| | then there exists a rainbow s t-path. − The complex in action here is the collection NREACH of sets of edges, that do not contain an s t path. It is not hard to show that λ¯(NREACH) n, which can be − ≤ combined with Theorem 3.5 to yield Theorem 7.1.

Conjecture 11.4. λ¯(NREACH) is the maximal number of vertices in a family of innerly disjoint s t paths. − Is there a cooperative version of Theorem 3.6? Somewhere over the rainbow there might well be one. Presently, only a few humble results exist in this direction. One of them concerns pairwise cooperation:

Theorem 11.5. [37]

18 (1) Let S ,...,S be subsets of Rd, and let ~v Rd. If for every 1 i < j d +1 we 1 d+1 ∈ ≤ ≤ have ~v conv(A A ) then there exists a rainbow set R such that ~v conv(R). ∈ i ∪ j ∈ d (2) Let S1,...,Sd be non-empty sets in R , all contained in the same closed half-space. Suppose that for every 1 i < j d we have v cone(A A ). Then there exists ≤ ≤ ∈ i ∪ j a rainbow set R for the sets A , such that v cone(R). i ∈ A similar extension exists for Theorem 2.4. For a system of sets of edges, denote by F ν ( ) the maximal size of a rainbow matching. r F Theorem 11.6. [7] Let =(F ,...,F ) be a system of bipartite sets of edges, sharing F 1 2k−1 the same bipartition. If each F is non-empty, and ν(F F ) k for every i < j 2k 1, i i ∪ j ≥ ≤ − then ν ( ) k. r F ≥ The proof of Theorem 11.6 uses a cooperative result on networks, which we omit. J. Kim [41] proved a cooperative version of Theorem 3.5:

Theorem 11.7. Let be a simplicial complex and let V = V ( ). Assume that V . Let C C 6∈ C =(S ,...,S ) be a system of subsets of V . Suppose that for each I [m] either S 1 m ⊆ SI 6∈ C or lk ( ) is m 1 I -Leray. Then has a rainbow set not belonging to . C SI − −| | S C (Here lk (S) is the link of S in .) Using this theorem it is possible to prove another C C cooperative generalization of Theorem 3.6

Theorem 11.8. Let t 1, If S ,...,S are subsets of Rd satisfy the condition that for ≥ 1 d+t every I [d + t] there holds ~v conv(S ) then there exists a rainbow set S such that ⊆ ∈ I ~v conv(S). ∈ Theorem 3.6 is the case t = 1.

11.1. Cooperative conditions for spanning rainbow sets. Here is a cooperative version of Theorem 6.3.

Theorem 11.9. Let be a matroid of rank n, let T V ( ), and let =(A ,...,A ) M ⊆ M A 1 n be a system of subsets of V ( ), satisfying the following: for every J [n], either rk(A ) M ⊆ J ≥ J or v span (A ). Then there exists an -rainbow set spanning T . | | ∈ M J A This is a common generalization of Rado’s theorem (Theorem 2.1), which is the case T = V , and Theorem 6.3 which is the case in which all Js satisfy the second condition.

19 Proof. As noted, we may assume that there exists J [n] for which rk( ) < J , or else ⊂ AJ | | by Rado’s theorem (Theorem 2.1) there exists a rainbow base. Choose a minimal such J. Certainly J rk( )+1 1, so choose any j J. Applying Rado’s theorem to | | ≥ AJ ≥ ∈ I := J j yields then a rainbow R representing sets A for all i I. By the \{ } ∈ M i ∈ assumption of the theorem T span( ), so it suﬃces to prove that span( ) = span(R). ⊆ AJ AJ Clearly span(R) span( ), and since rk(R) = rk( )= J 1, this inclusion is in fact ⊆ AJ AJ | | − equality.

This yields a cooperative version of Theorem 6.2. Denote by (G) is the set of compo- C nents of a graph G.

Theorem 11.10. Let F1,...,Fn be sets of edges in a graph with n vertices, satisfying the following condition: for every J [n] either ⊆ ( V (C) 1) J , or • PC∈C(FJ ) | | − ≥| | F contains an odd cycle. • J Then there exists a rainbow odd cycle.

12. Rainbow independent sets in hypergraphs

We conclude with a rather neglected topic: rainbow independent sets in hypergraphs (to be distinguished from rainbow matchings in hypergraphs, which are independent sets in line graphs). A set I of vertices is said to be independent in a hypergraph H if it does not contain any edge of H. By α(H) we denote the largest size of an independent set. In [24] and in [48] the following was proved:

|V (H)| Theorem 12.1. α(H) 1 −1 ≥ d /k Using the Lov´asz Local Lemma it is possible to prove:

Theorem 12.2. If H is k-uniform with maximal vertex degree d, then any collection of disjoint sets of vertices of H, each of size at least ked1/k−1, has a full rainbow independent set.

But the coeﬃcient ked can probably be replaced by a smaller one, for example by a result of Haxell [34], for k = 2 (namely for graphs) the coeﬃcient is 2d, rather than 2ed. Here is a conjecture for k = 3, stemming from examples whose description we omit:

20 Conjecture 12.3. If H is 3-uniform with maximal degree d, then any collection of disjoint sets of vertices of H, each of size at least 1.1√d, has a full rainbow independent set.

Using the Lov´asz Local Lemma it is possible to prove this result with 1.1√d replaced by 3e√d. Data Availability No data was needed to produce the results in this paper. Acknowledgements We are indebted for fruitful discussions with Eli Berger, Ron Holz- man and Zilin Jiang. We are also indebted to Dmitry Falikman for computer tests of some of the conjectures, and useful insights.

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