Almost all hypergraphs without Fano planes are bipartite∗
Yury Person† Mathias Schacht‡
Abstract hypergraph H turns out not to be bipartite. More- The hypergraph of the Fano plane is the unique 3-uniform over, we prove that only with even smaller probability, −Ω(n3) hypergraph with 7 triples on 7 vertices in which every pair 2 , the Fano-free hypergraphs are “far” from being of vertices is contained in a unique triple. This hypergraph bipartite (see Lemma 4.1 for the precise statement). is not 2-colorable, but becomes so on deleting any hyperedge Similar questions for graphs (following the work from it. We show that taking uniformly at random a labeled of Kleitman and Rothschild [13] on posets) were first 3-uniform hypergraph H on n vertices not containing the studied by Erd˝os,Kleitman, and Rothschild [7], who hypergraph of the Fano plane, H turns out to be 2-colorable were interested in Forb(n, L) for a fixed graph L. In 2 with probability at least 1 − 2−Ω(n ). For the proof of particular, those authors proved that almost every this result we will study structural properties of Fano-free triangle-free graph is bipartite. Moreover, they showed hypergraphs. for p ≥ 3
(1− 1 )(n)+o(n2) 1 Introduction (1.1) Forb(n, Kp+1) ≤ 2 p 2 . We study monotone properties of the type Forb(n, L) Thus, the dominating term in the exponent turns out to for a fixed hypergraph L, i.e., the family of all labeled be the extremal number ex(n, Kp+1), where by ex(n, L) hypergraphs on n vertices, which contain no copy we mean the maximum number of edges a graph on n of L as a (not necessarily induced) subgraph. We vertices can have without containing L as a subgraph. obtain structural results for an “average” member of Later, Erd˝os, Frankl, and R¨odl [6] extended this result Forb(n, F ), where F is the 3-uniform hypergraph of the from cliques to arbitrary graphs L with chromatic Fano plane, which is the unique triple system with 7 number, χ(L), at least three, by proving hyperedges on 7 vertices where every pair of vertices is contained in precisely one hyperedge. The hypergraph (1.2) |Forb(n, L)| ≤ 2(1+o(1))ex(n,L). of the Fano plane F is not 2-colorable, i.e., for every vertex partition X∪˙ Y = V (F ) into two classes there A strengthening of (1.1) was obtained by Kolaitis, exists an edge of F which is either contained in X or Pr¨omel, and Rothschild [15, 16], who showed that in Y . Consequently, Forb(n, F ) contains any 2-colorable almost every Kp+1-free graph is p-colorable. This result 3-uniform hypergraph on n vertices. was further extended by Pr¨omeland Steger [20] (see Roughly speaking, we show that almost every hy- also [10]) from cliques to such graphs L, which contain pergraph H ∈ Forb(n, F ) is 2-colorable. More precisely, a color-critical edge, i.e., an edge e ∈ E(L) such that let Bn be the class of all labeled 2-colorable (or bipar- χ(L − e) < χ(L) with L − e = (V (L),E(L) \{e}). The tite) hypergraphs on n vertices. We prove that result of Pr¨omel and Steger states that for graphs L with χ(L) = p + 1 ≥ 3 almost every L-free graph is p- |Forb(n, F )| 2 ≤ (1 + 2−Ω(n )) colorable if and only if L contains a color-critical edge, |Bn| which was conjectured earlier by Simonovits [3]. Recently, Balogh, Bollob´as, and Simonovits [2] (see Theorem 1.1). This shows that choosing uniformly showed a sharper version of (1.2): at random a 3-uniform Fano-free hypergraph H, then −Ω(n2) only with exponentially small probability, 2 , the (1− 1 ) n +O(n2−γ ) |Forb(n, L)| ≤ 2 p (2) ,
∗Supported by GIF grant no. I-889-182.6/2005. where p = χ(L) − 1 and γ = γ(L) > 0 is some constant †Institut f¨ur Informatik, Humboldt-Universit¨at zu depending on L (see also [1] for more structural results Berlin, Unter den Linden 6, D-10099 Berlin, Germany, by the same authors). [email protected]. ‡Institut f¨ur Informatik, Humboldt-Universit¨at zu In this paper we will study this type of questions Berlin, Unter den Linden 6, D-10099 Berlin, Germany, for k-uniform hypergraphs, where by a k-uniform hyper- V [email protected]. graph H we mean a pair (V,E) with E ⊆ k . Let L be a k-uniform hypergraph and denote by Forb(n, L) Theorem 1.1. Let F be the 3-uniform hypergraph of the set of all labeled k-uniform hypergraphs H on n the Fano plane. There exist a real c > 0 and an vertices not containing a (not necessarily induced) copy integer n0, such that for every n ≥ n0 we have of L. Note, that the main term in the exponent in the −cn2 graph case was ex(n, L). Recently, in [17, 18] similar |Forb(n, F )| ≤ |Bn|(1 + 2 ). results were obtained for k-uniform hypergraphs, i.e, This result can be seen as a first attempt to derive k |Forb(n, L)| ≤ 2ex(n,L)+o(n ) results for hypergraphs in the spirit of Kolaitis, Pr¨omel and Rothschild [16] and Pr¨omel and Steger [20]. We for arbitrary k-uniform hypergraphs L. will prove Theorem 1.1 using techniques developed Furthermore, we say a k-uniform hypergraph H by Balogh, Bollob´asand Simonovits in [2]. In fact, contains a (p + 1)-color-critical edge, if there exists a with these methods, one could also reprove the above hyperedge e in H such that after its deletion one can mentioned theorems for graphs containing color-critical color the vertices of H with p colors without creating a edges [3]. monochromatic edge in H0 = (V (H),E(H) \{e}), but one still needs p + 1 colors to color H. 2 Notation and Outline In this paper we will sharpen the bound on From now on we will consider only 3-uniform hyper- |Forb(n, L)| in the special case, when L is the 3-uniform graphs and by a hypergraph we will always mean a 3- hypergraph of the Fano plane. It was shown indepen- uniform hypergraph. For the sake of a simpler notation dently by F¨urediand Simonovits [9] and Keevash and we set Sudakov [12], that the unique extremal Fano-free hyper- Fn = Forb(n, F ) , graph is the balanced, complete, bipartite hypergraph where by F we will always denote the hypergraph of the Bn = (U∪˙ W, EB ), where |U| = bn/2c, |W | = dn/2e n Fano plane. We will refer to hypergraphs not containing and EBn consists of all hyperedges with at least one vertex in U and one vertex in W . Therefore, for the a copy of F , as Fano-free hypergraphs. hypergraph of the Fano plane F we have We will use the following estimates (often without mentioning them explicitly). First of all we note that
ex(n, F ) = e(Bn) = |EBn | we can bound |Bn| by n dn/2e bn/2c e(Bn) n e(Bn) = − − (2.4) 2 ≤ |Bn| ≤ 2 · 2 , 3 3 3 n n3 n2 n3 as there are at most 2 partitions of [n] in two disjoint = − − O(n) ≤ 8 4 8 sets and there are at most e(Bn) hyperedges running between those two sets. and For a given hypergraph H = (V,E) a vertex v ∈ V lnm jnk bn/2c 3 we define its link LH (v) = (V \{v},Ev) to be the graph (1.3) δ (B ) = − 1 + ≥ n2 −n , 1 n 2 2 2 8 whose edges together with v form hyperedges of H, namely where for a hypergraph H = (V,E) we denote by δ1(H) Ev = {{u, w}: {v, u, w} ∈ E} . the minimum vertex degree, i.e., We define the degree of v ∈ V to be deg(v) = degH (v) = δ1(H) = min {v, w}: {u, v, w} ∈ E . |E(LH (v))|, and sometimes we omit H when it is clear u∈V from the context. For A ⊆ V (H) and v ∈ V (H) denote Furthermore, we have by A LA(v) = LH (x)[A] = (A \{v},Ev ∩ 2 ) e(B ) = e(B ) + δ (B ) + δ (B ) + δ (B ). n n−3 1 n 1 n−1 1 n−2 the link of v induced on A. Note that the hypergraph of the Fano plane is a 3-color- By h(x) = −x log x − (1 − x) log(1 − x) we denote critical hypergraph, i.e., the hypergraph of the Fano the entropy function, and by log we always mean log2. plane becomes 2-colorable after deletion of any edge. Note further, h(x) → 0 as x tends to 0, and h(x) ≥ x for 1 Let Bn denote the set of all labeled bipartite 3- x ≤ 2 . We will use the entropy function h(x) together n h(x)n uniform hypergraphs on n vertices. We know that every with the well-known inequality xn ≤ 2 , which labeled bipartite hypergraph does not contain a copy of holds for 0 < x < 1. Furthermore, we will use that for P n n the hypergraph of the Fano plane. On the other hand, n > 3k we have j
0 3 γ Y Y Fn(α) = H ∈ Fn : e(XH ) + e(YH ) < αn (3.5) |{L ⊆ H}| ≥ (1 − 2 ) de |Vi| p e∈E(L) i∈[`] and |XH |, |YH | < n/2 + 2 h(6α)n . X 0 1 + (1E(e0(L−)) − de0 ) . 0 Lemma 4.1. For every α ∈ (0, 12 ) there exist c > 0 L0 ⊆H 0 0 − and an integer n0 such that for all n ≥ n0 0 0 3 P 0 e(Bn)−c n We bound the error term 0 (1E(e0(L )) − de0 ) L−⊆H − |Fn \Fn(α)| < 2 . from below. For that, we will appeal to the one-sided Proof. The proof of Lemma 4.1 combines the weak hy- regularity of (Vi)i∈e0 . Let L∗ = L[[`]\e0] be the induced subhypergraph of L obtained by removing the vertices pergraph regularity lemma with the stability theorem 0 for Fano-free hypergraphs applied to the reduced hy- of e0 and all edges of L intersecting e0. For a copy L∗ of L in H, let ext(L0 ) be the set of triples T ∈ Q V pergraph. ∗ ∗ i∈e0 i such that V (L0 )∪˙ T spans a copy of L0 in H. Hence, Let λ = λ(α/2) be given by Theorem 3.1. We may ∗ − assume λ < 16h(6α). We set X 0 X X (1E(e0(L )) − de ) = (1E(T ) − de ) λ − 0 0 c0 = . 0 L0 ⊆H 0 L−⊆H ∗ T ∈ext(L∗) 17 and, moreover, since L is a linear hypergraph, we have We choose η such that λ > 16h(6η) and η ≤ α/2. Finally let ε = ε(η/2) ≤ η/2 be given by Theorem 3.3. |e0 ∩ e| ≤ 1 for every edge e of L−. Hence, for every fixed copy L0 of L in H and every i ∈ e , there exists Set t0 = 1/ε and let n be sufficiently large, in particular, ∗ ∗ 0 0 L0 set n max{T , n }, where T and n are given by a subset W ∗ ⊆ V such that 0 0 0 0 0 i i the weak regularity lemma, Theorem 3.2. For the main 0 0 Y L∗ steps of the proof it is sufficient to keep in mind that (3.6) ext(L∗) = Wi . −1 i∈e0 0 < ε = t0 ≤ η λ α.
0 L∗ We may assume in the following that t divides n, and Indeed, for every i ∈ e0, the set Wi consists of those 0 thus |Vi| = n/t, i = 1, . . . , t, as this does not affect our vertices v ∈ Vi with the property that V (L∗)∪{˙ v} spans a copy of L induced on V (L )∪{˙ i} in H. Therefore, we asymptotic considerations. ∗ 0 can bound the error term as follows We will upper bound |Fn \Fn(α)| in two steps. In the first step we bound the number of hypergraphs H 3 X 0 that have e(XH ) + e(YH ) ≥ αn . In the second (1E(e0(L−)) − de0 ) L0 ⊆H step we show that most of the hypergraphs H with − 3 e(XH ) + e(YH ) < αn will have nearly equal sizes: X X n Y L0 o = 1 (T ) − d : T ∈ W ∗ E e0 i n p L0 ⊆H i∈e max{|XH |, |YH |} < + 2 h(6α)n . ∗ 0 2 X Y γ Y Y ≥ − ε |Vi| ≥ − de |Vi| , 2 Step 1. Consider a hypergraph H ∈ Fn satisfying 0 i∈e L∗⊆H 0 e∈E(L) i∈[`] 3 e(XH ) + e(YH ) ≥ αn . We apply the weak regularity lemma, Theorem 3.2, with parameters ε and t . Firstly, where the one-sided (ε, de0 )-regularity, the choice of ε, 0 and (3.6) were used for the last two estimates. Now the we estimate the number of hyperedges, which are con- proposition follows from (3.5), which implies immedi- tained in the “uncontrolled” part of the regular parti- ately Theorem 3.3. tion: • the number of hyperedges intersecting at most two 4 Proof of the main result of the clusters is at most 4.1 Almost bipartite hypergraphs. Our first step n/t 1 for the proof of Theorem 1.1 is an estimate on the t n < n3, 2 2t number of those hypergraphs H ∈ Fn which are far from 3 being bipartite, namely for which e(XH )+e(YH ) ≥ αn • the number of hyperedges contained in irregular for some α > 0 to be specified later. Thus, the triples is at most remaining hypergraphs will admit a “nice” partition. t n3 ε Moreover, most of these remaining hypergraphs H will ε < n3, have partition classes of nearly same size. 3 t 6 • the number of hyperedges that are contained in ε- Note that there are at most 2n possible partitions, and regular triples of density less than η is at most since less than αn3 hyperedges are completely contained in X and Y , those hyperedges can be chosen in at n3 t η H H η < n3. most 3 t 3 6 αn −1 n n X 3 ≤ 3 3 i αn3 Thus, the number of discarded edges is less than ηn . i=0 Secondly, consider the resulting cluster-hypergraph ways. Finally, as we assumed that our partitions H(η). It must be Fano-free as otherwise Theorem 3.3 are “unbalanced” we estimate the number of possi- would imply that H also contains a copy of the hyper- ble choices of hyperedges between XH and YH by 3 3 graph of the Fano plane. We assumed that e(XH ) + 2n /8−2h(6α)n . Altogether we get, that there are at 3 e(YH ) ≥ αn , so we can bound the number of hyper- most 3 edges in H(η) from above by (1 − λ)t /8. Otherwise, n 3 3 3 3 3 Theorem 3.1 would give us a partition of V ,...,V into 2n · 3 · 2n /8−2h(6α)n ≤ 2n+h(6α)n /6+n /8−2h(6α)n 1 t αn3 disjoint sets X and Y with e (X) + e (Y ) < H(η) H(η) 3 3 αt3/2. Defining a partition of V (H) into the following ≤ 2n /8−h(6α)n two sets 3 hypergraphs with e(XH ) + e(YH ) < αn and [ [ A = U and B = W, n p max{|XH |, |YH |} ≥ + 2 h(6α)n. U∈X W ∈Y 2 with Combining Step 1 and 2 we obtain 3 3 3 3 α n3 |F \F 0 (α)| ≤ 2n /8−λn /16 + 2n /8−h(6α)n e (A) + e (B) < ηn3 + t3 ≤ αn3 , n n H H 3 3 2 t < 2n /8−λn /16+1, 3 which yields a contradiction to e(XH ) + e(YH ) ≥ αn . 3 2 since h(6α) > λ/16. Due to n /8−e(Bn) ≤ n /4+O(n) Now we are able to bound the number of hyper- 0 3 and the choice of c = λ/17, the lemma follows for graphs H ∈ Fn with e(XH ) + e(YH ) ≥ αn from above sufficiently large n. by calculating the total possible number of ε-regular partitions together with all possible cluster-hypergraphs 4.2 Everywhere dense hypergraphs. Now we associated with them and all possible hypergraphs that know that almost all Fano-free hypergraphs are nearly could give rise to such a particular cluster-hypergraph. bipartite and admit a partition into almost equal classes. This way we get We want to restrict our consideration to those hyper- 3 graphs that in addition have no sparse “bipartite” spots. Fn \{H ∈ Fn : e(XH ) + e(YH ) ≥ αn } 3 Our motivation comes from random bipartite hyper- T0 ηn −1 3 n 1 N X n t (1−λ) t ( n )3 X graphs. Namely, we would expect N edges having ≤ t · 2(3) · 2 8 t · 3 2 2 j exactly one end in the first class and two in the sec- t=t0 j=0 ond in H(N,N, 1/2), the random bipartite hypergraph n n+1 T0 (1−λ)n3/8 with both classes of size N where each edge exists with ≤ T · 2( 3 ) · 2 · 3 0 ηn3 probability 1/2. If we would take any three disjoint sub- (n+1) log T + T0 +n3/8−λn3/8+h(6η)n3/6 sets not all of them in one partition class, each of size, ≤ 2 0 ( 3 ) say m, then we would expect there m3/2 hyperedges. 3 3 < 2n /8−λn /16 , Deviations from this value, say only m3/4 edges instead of m3/2, would only happen with very small probabil- for sufficiently large n, due to the choice of λ. ity. The following lemma, Lemma 4.2, states that this intuition holds for almost all hypergraphs in F 0 (α). Step 2. We now estimate the number of those hy- n 00 3 Definition 4.2. Let F (α, β) denote the family of pergraphs H, for which e(XH ) + e(YH ) < αn , but n p those hypergraphs H ∈ F 0 (α), for which the following max{|XH |, |YH |} ≥ n/2 + 2 h(6α)n. First we upper n condition holds. bound e(XH ,YH ) for such a hypergraph H by For any pairwise disjoint sets W ⊂ X , W ⊂ Y 1 H 2 H |YH | |XH | and W ⊂ Z , where Z ∈ {X ,Y }, with |W | ≥ βn e(X ,Y ) ≤ |X | + |Y | 3 H H H H i H H H 2 H 2 for i = 1, 2, 3 we have n n3 1 < |X ||Y | < − 2h(6α)n3. e (W ,W ,W ) ≥ |W ||W ||W |. 2 H H 8 H 1 2 3 4 1 2 3 The following lemma shows that most hypergraphs in For the proof of Lemma 4.3, we will use a simple 0 00 Fn(α) belong to Fn (α, β). consequence of the regularity lemma for graphs. For that we need the definition of an ε-regular pair in Lemma 4.2. For every β > 0 there exist α, c00 > 0 and 00 00 graphs. an integer n0 such that for all n ≥ n0 For a bipartite graph G = (V1,V2,E) we define the 00 3 |E| 0 00 e(Bn)−c n density of G as d(V1,V2) = . The density of a |Fn(α) \Fn (α, β)| < 2 . |V1||V2| subpair (U1,U2), where Ui ⊆ Vi for i = 1, 2 is defined Proof. Choose α > 0 such that e(U1,U2) as d(U1,U2) = . Here e(U1,U2) denotes the |U1||U2| 3 β (1 − h (1/4)) ≥ h(6α)/3 , number of edges with one vertex in U1 and one vertex in U2. We say, G, or simply (V1,V2), is ε-regular if all set c00 = h(6α)/7, and let n00 be sufficiently large. 0 subsets Ui ⊆ Vi, with |Ui| ≥ ε|Vi|, i = 1, 2, satisfy Below we bound the number of hypergraphs H with 0 00 n H ∈ Fn(α) \Fn (α, β). There are at most 2 partitions |d(V1,V2) − d(U1,U2)| ≤ ε. XH ∪˙ YH = [n] of the vertex set and we can choose the edges lying completely within XH and YH in at most Now we state the theorem, which one could easily obtain from the usual regularity lemma [23] (for a better 3 αn −1 n n X dependency of the constants one also could also use [19, 3 ≤ 3 j αn3 Theorem 1.1]). j=0 Theorem 4.1. For every γ > 0 and ε ∈ (0, γ/3) there possible ways. A simple averaging argument shows that exist T ,N such that the following holds. it suffices to consider sets W with |W | = βn and there 0 0 i i For all vertex disjoint graphs G and G on are at most X Y |V (GX )| + |V (GY )| = n ≥ N0 vertices with e(GX ), 2 p 3 e(GY ) ≥ γn there exist t ≤ T0 and pairwise dis- n/2 + 2 h(6α)n X β3n3 2 joint sets X1,X2,Y1,Y2,Y3,Y4, each of size n/t, and βn i 0≤i<β3n3/4 X1,X2 ⊂ V (GX ) and Yi ⊂ V (GY ), i ∈ [4], so that β3n3 GX [X1,X2], GY [Y1,Y2] and GY [Y3,Y4] are ε-regular < 23n+1 with density at least γ/3. β3n3/4 With this result at hand we can give the proof of ways to select W1,W2,W3 and the hyperedges in Lemma 4.3. eH (W1,W2,W3). Finally, there are at most 1 3 3 Proof. Let ε = min{ ε(γ/6), γ/6}, where ε(γ/6) is 2e(Bn)−β n 2 given by Theorem 3.3. Set β = ε/(2T0), with T0 = ways to choose the remaining edges of H. Multiplying T0(γ, ε) given by Theorem 4.1. Let α = α(β) be given everything together, we obtain by Lemma 4.2 and let n0 be sufficiently large. Again, it is sufficient to keep in mind: 0 00 |Fn(α) \Fn (α, β)| −1 n 3 3 0 α β T ε γ. β n 3 3 0 4n+1 3 e(Bn)−β n ≤ 2 3 3 3 2 αn β n /4 We prove our lemma by contradiction. More pre- 3 3 3 3 3 ≤ 24n+1+h(6α)n /6+h(1/4)β n +e(Bn)−β n cisely, we will assume that there exists a hypergraph 00 2 00 3 H ∈ F (α, β) with max{∆(H[X ]), ∆(H[Y ])} ≥ γn , e(Bn)−c n n H H ≤ 2 , and we will show that H contains a copy of the hyper- for sufficiently large n. graph of the Fano plane. Without loss of generality assume that there exists 00 We will also need the following useful observation, H ∈ Fn (α, β) and a vertex x ∈ XH with degH[X](x) ≥ 00 2 2 that for suitably chosen α and β, every H ∈ Fn (α, β) γn . Thus, e(LY (x)) ≥ e(LX (x)) ≥ γn , as otherwise has no vertex of high degree in its own partition class. this violates the minimality condition of the partition X ∪˙ Y = V (H). We consider the graphs Lemma 4.3. For every γ > 0 there exist α, β > 0 and H H 00 an integer n0, such that for every H ∈ Fn (α, β) we have X GX = LX (x) = (XH \{x},Ex ∩ 2 ) 2 max{∆(H[XH ]), ∆(H[YH ])} < γn , and Y GY = LY (x) = (YH ,Ex ∩ ) for all n ≥ n0. 2 ˙ 1 6 m and apply Theorem 4.1 to GX ∪GY . This way we obtain where E(X) = ( 8 ) 4 is the expectation and ε-regular pairs (X ,X ) ⊂ G and (Y ,Y ), (Y ,Y ) ⊂ 1 2 X 1 2 3 4 GY , with |Xi| = |Yj| ≥ (n − 1)/T0 and i ∈ [2], j ∈ [4], X Y each of density at least γ/3. ∆ = E tj Consider the following 7-partite subhypergraph L A,B∈A:A∩B6=∅ j∈A∪B X Y with vertex classes {x}, X1, X2, Y1, Y2, Y3, and Y4. ≤ (X) sup t x E E j Denote L to be the hypergraph obtained from L by A∈A B∈A:A∩B6=∅ blowing up its first vertex class {x} to the size of X (all 1 5 3! other partition classes are equal), and denote this m 1 1 ≤ E(X) · 6 · + m · 4 · blown-up class by X˜. More precisely, Lx = (W x,Ex), 2 8 8 where 3m2 + 253m m2 ≤ (X) ≤ (X) . x E 15 E 13 W = X˜∪˙ X1∪˙ X2∪˙ Y1∪˙ Y2∪˙ Y3∪˙ Y4 2 2 Hence, we obtain and 12 m ! 2 E(X) 4 Pr(X = 0) ≤ exp − 2 = exp − 6 2 {a, b, c} ∈ Ex m 2 m ( −11 2 {a, b, c} ∈ E(L) , if {a, b, c} ∩ X˜ = ∅ , ≤ exp −2 m . ⇔ {x, b, c} ∈ E(L) , if a ∈ X˜ and b, c 6∈ X.˜ We finally define the last subclass of Fano-free hypergraphs. Note, that L contains a copy of the hypergraph of Definition 4.3. Let F 000(α, β) denote the family of the Fano plane if, and only if, Lx contains one. Now n those hypergraphs H ∈ F 00(α, β), for which the following we apply Theorem 3.3 to Lx, as Lx contains now 7 one- n condition holds. sided (ε, γ/6)-regular triples and these triples form a For all triples z , z , z ∈ Z of vertices with Z ∈ Fano plane. This is true since the triples (X,X˜ ,X ), 1 2 3 1 2 {X ,Y } we have L (z ) ∩ L (z ) ∩ L (z ) ⊇ K , (X,Y˜ ,Y ) and (X,Y˜ ,Y ) “inherit” the ε-regularity H H Q 1 Q 2 Q 3 4 1 2 3 4 where {Q, Z} = {X ,Y }. In other words, we require from the ε-regular pairs of (X ,X ), (Y ,Y ), and H H 1 2 1 2 that the common link of any triple from X or Y (Y ,Y ), while the other triples are one-sided (ε, γ/6)- H H 3 4 contains a copy of K in the other vertex class. regular due to the choice of β and the properties of H ∈ 4 00 Fn (α, β). This yields a contradiction and Lemma 4.3 It follows directly from the definition, that every 000 000 follows. H ∈ Fn (α, β) is bipartite, i.e., Fn (α, β) ⊆ Bn. Otherwise, any hyperedge e, say in XH , together with 4.3 Proof of Theorem 1.1. We will need the fol- the K4 in YH , which lies in the common link of the lowing consequence from Janson’s inequality [11]. vertices of e would span a copy of the hypergraph of the Fano plane. We also note that we could replace K4 in 000 Lemma 4.4. The probability that the binomial random the definition of Fn (α, β) by a 1-factor of K4 that is 1 graph G(m, 8 ) with m ≥ 253 vertices and edge probabil- created by the union of the links of any three vertices. ity 1/8 does not contain a copy of K4 is bounded from We are now going to prove our main theorem, above by exp(−2−11m2). Theorem 1.1, by induction. The proof is based on the lemmas from the previous sections.
Proof. Let t1, . . . , t m be jointly independent Boolean Proof. (Theorem 1.1) We set ( 2 ) random variables representing the edges of G(m, 1 ). Let 8 ϑ [m] (4.7) ϑ = 2−17 and c = the collection of those 6-sets of the set 2 that corre- 3 spond to K4’s be denoted by A. Therefore, the random P Q and choose γ > 0 such that variable X = A∈A j∈A tj counts the number of K4’s 1 in G(m, 8 ). Applying Janson’s inequality [11] we bound (4.8) 3h(2γ) < ϑ . the probability of the event X = 0 by Let α and β > 0 be given by Lemma 4.3. We may also (X)2 assume that Pr(X = 0) ≤ exp −E , 2∆ (4.9) 3ph(6α) + 6h(6α) < ϑ/2 , as choosing α smaller we will only have to eventually in its own partition class. This again bounds the number increase n0. Again, it is sufficient to keep in mind that of ways for choosing these hyperedges by
2 3 −17 γn −1 n n 3 0 < α β γ ϑ = 2 . X 2 ≤ 2 . j γn2 Let c0 and c00 be given by Lemma 4.1 and Lemma 4.2. j=0
Finally, let 2 For every vertex in S we have at most 2n /4 possibilities 20 0 00 for choosing edges with one more end in the same n0 ≥ max{2 , 14/ϑ, 1/c , 1/c } partition as S and the other end in the other partition be sufficiently large so that Lemma 4.1, Lemma 4.2 and class, this gives us at most Lemma 4.3 hold. 2 23n /4 By induction on n we will verify the following statement, which implies Theorem 1.1 ways to choose that type of hyperedges. The last estimate concerns the number of ways we can connect n2n−cn2 (4.10) |Fn| ≤ |Bn|(1 + 2 0 ) . our triple S to the other partition class, say Y , without creating any single copy of K4, which is contained in the n2n−cn2 For n ≤ n0 the statement is trivial, since then 2 0 joint link of the vertices from S. Here we use Lemma 4.4. is bigger than the number of all hypergraphs on n For every vertex v in S, say S ⊂ X, we can choose its (|Y |) vertices and we now proceed with the induction step link graph LY (v) in at most 2 2 ways. However, since and verify (4.10) for n > n0. the joint link of three vertices in S contains no K4, we The proof is based on the following chains infer from Lemma 4.4, that there are at most
0 00 000 3(|Y |) −11 2 3(|Y |)−|Y |2/211 Fn ⊇ Fn(α) ⊇ Fn (α, β) ⊇ Fn (α, β), 2 2 exp(−2 |Y | ) < 2 2 and ways to choose all three link graphs such that no K4 F ⊇ B ⊇ F 000(α, β). appears in the joint link. n n n Combining the above estimates and Consequently, n/4 ≤ |Y | ≤ n/2 + 2ph(6α)n , 0 0 00 |Fn| ≤ |Fn \Fn(α)| + |Fn(α) \Fn (α, β)| we obtain 00 000 + |Fn (α, β) \Fn (α, β)| + |Bn| . 00 000 |Fn (α, β) \Fn (α, β)| 0 n 3 Lemma 4.1 bounds |F \F (α)| and Lemma 4.2 bounds |Y | 2 11 n n n n−3 2 3n2/4 3( )−|Y | /2 0 00 ≤ 2 · 2 2 2 |Fn−3| |Fn(α) \Fn (α, β)|. Hence, it remains to estimate 3 γn2 00 000 |Fn (α, β) \Fn (α, β)|. (4.9) 3 log n+n+3h(2γ)n2/2+9n2/8+ϑn2/2−n2/215 For that we will use the induction assumption and ≤ 2 |Fn−3| 00 000 proceed as follows. Let H ∈ F (α, β) \F (α, β) and (4.8) 2 2 16 n n δ1(Bn−2)+δ1(Bn−1)+δ1(Bn)+ϑn −n /2 ≤ 2 |Fn−3| XH ∪˙ YH be its minimal partition. Consider a subset S ∈ XH YH (4.7) 2 ˙ δ1(Bn−2)+δ1(Bn−1)+δ1(Bn)−ϑn 3 ∪ 3 . Deleting S from H, we obtain a Fano-free = 2 |Fn−3| hypergraph H0 on n−3 vertices, where V (H0) = [n]\S. (4.10) 2 2 2 00 000 −ϑn n−3 n0(n−3)−c(n−3) On the other hand, for every H ∈ Fn (α, β) \Fn (α, β) ≤ 2 · 2 · |Bn| · (1 + 2 ) 0 there exists a hypergraph H ∈ Fn−3 such that H can −ϑn2/2 n2n−cn2−ϑn2/2 ≤ |B |(2 + 2 0 ) , be reconstructed from H0 in the following way. For n 0 H ∈ Fn−3 we choose a set S of 3 vertices, which where we used (2.4) for the penultimate inequality and we “connect” in an appropriate manner, so that the n0 ≥ 14/ϑ and c ≤ 1 for the last inequality. Finally, we 00 000 resulting hypergraph is in Fn (α, β) \Fn (α, β). derive the required upper bound on |Fn| We can choose the set S, the partition of H0 and |F | ≤ |F \F 0 (α)| + |F 0 (α) \F 00(α, β)| the set which contains S in at most n n n n n + |F 00(α, β) \F 000(α, β)| + |B | n n n n n−3 0 3 00 3 2 ≤ 2e(Bn)−c n + 2e(Bn)−c n 3 −ϑn2/2 n2n−cn2−ϑn2/2 + |Bn|(2 + 2 0 ) + |Bn| ways. 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