Sunflowers and $ L $-Intersecting Families

arXiv:1601.04897v1 [math.CO] 19 Jan 2016 [ o ah1 each for e [ Let Introduction 1 with Keywords. n X k y2 by ] ecl family a call We family A Let h aiyo all of family the n m k than e ytmo oethan more of system set petals. tn o h set the for stand ] -uniform, uflwr and Sunflowers X [ easif petals n Let egv loa pe on for bound upper an also give We egv nuprbudfor bound upper an give We Let ] . ≤ eafie usto [ of subset fixed a be f ( g f F ,r s r, k, ,j i, ∆ ( ( ,r ℓ r, k, ,r s r, k, -system, = L ≤ itretn e ytm where system, set -intersecting lmns then elements, ) { m F etelatnme ota any that so number least the be ) tn o h es ubrs htif that so number least the for stand ) F 1 . k F , . . . , fsbeso [ of subsets of lmn ust of subsets element { L 1 , itretn aiis xrmlsttheory set extremal families, -intersecting coe 1 2016 11, October 2 m Gábor Heged˝us n , . . . , bd University Obuda ´ } F g fsbeso [ of subsets of ( i n ,r ℓ r, k, L ∩ .Fra nee 0 integer an For ]. Abstract F F itretn families -intersecting } f otisasnoe with sunflower a contains n j ednt h aiyo l ust of subsets all of family the denote We . ( escnan uflwrwith sunflower a contains sets ) 1 ] = k, k -uniform 3 g t \ =1 m s , ( ,r ℓ r, k, X ). F . t n sa is ] ). | L if , | k = -uniform, sunflower | s F ≤ and , | F k = sa arbitrary an is ≤ k F ℓ o each for r n -intersecting a more has o ∆ (or petals. ednt by denote we -system F r F ∈ ) . The intersection of the members of a sunflower form its kernel. Clearly a a family of disjoint sets is a sunflower with empty kernel. Erd˝os and Rado gave an upper bound for the size of a k-uniform family without a sunflower with r petals in [6]. Theorem 1.1 (Sunflower theorem) If is a k-uniform set system with more than F k 1 − t k!(r 1)k(1 ) − − (t + 1)!(r 1)t Xt=1 − members, then contains a sunflower with r petals. F Kostochka improved this upper bound in [9]. Theorem 1.2 Let r > 2 and α > 1 be fixed integers. Let k be an arbitrary integer. Then there exists a constant D(r, α) such that if is a k-uniform set system with more than F

(log log log k)2 k D(r, α)k! α log log k members, then contains a sunﬂower with r petals. F Erd˝os and Rado gave in [6] a construction of a k-uniform set system with (r 1)k members such that does not contain any sunﬂower with r petals. Later− Abbott, Hanson and SauerF improved this construction in [1] and proved the following result.

Theorem 1.3 There exists a c > 0 positive constant and a k-uniform set system such that F k/2 c log k > 2 10 − |F| · and does not contain any sunﬂower with 3 petals. F Erd˝os and Rado conjectured also the following statement in [6]. Conjecture 1 For each r, there exists a constant C such that if is a r F k-uniform set system with more than

k Cr members, then contains a sunflower with r petals. F 2 Erd˝os has offered 1000 dollars for the proof or disproof of this conjecture for r = 3 (see [4]). We prove here Conjecture 1 in the case of some special L-intersecting and ℓ-intersecting families. A family is ℓ-intersecting, if F F ′ ℓ whenever F, F ′ . Specially, F | ∩ | ≥ ∈F is an intersecting family, if F F ′ = whenever F, F ′ . F Erd˝os, Ko and Rado proved the∩ following6 ∅ well-known∈F result in [5]: Theorem 1.4 Let n, k, t be integers with 0 n0(k, t), n t − . |F| ≤ k t − n t [n] Further, = − if and only if for some T we have |F| k t ∈ t − [n] = F : T F . F { ∈ k ⊆ } Let L be a set of nonnegative integers. A family is L-intersecting, if F E F L for every pair E, F of distinct members of . In this terminology a| k∩-uniform| ∈ set system is a t-intersecting family iffF it is an L-intersecting family, whereFL = t, t +1,...,k 1 . { − } The following result gives a remarkable upper bound for the size of a k-uniform L-intersecting family (see [11]).

Theorem 1.5 (Ray-Chaudhuri–Wilson) Let 0 < s k n be positive integers. Let L be a set of s nonnegative integers and≤ an≤L-intersecting, k-uniform family of subsets of [n]. Then F n . |F| ≤ s Deza proved the following result in [3]. Theorem 1.6 (Deza) Let λ> 0 be a positive integer. Let L := λ . If is an L-intersecting, k-uniform family of subsets of [n], then either{ } F k2 k +1 |F| ≤ − or is a sunﬂower, i.e. all the pairwise intersections are the same set with λ elements.F

3 We generalize Theorem 1.6 for L-intersecting families in the following.

Theorem 1.7 Let be a family of subsets of [n] such that does not F F contain any sunﬂowers with three petals. Let L = ℓ1 <...<ℓs be a set of s non-negative integers. Suppose that is a k{-uniform, L-intersecting} family. Then F 5 2 (s 1) (1+ √ )k(s 1) (k k + 2)8 − 2 5 − . |F| ≤ − Next we improve Theorem 1.1 in the case of ℓ-intersecting families.

Theorem 1.8 Let r > 2 and α > 1 be ﬁxed integers. Let be an ℓ- F intersecting, k-uniform family of subsets of [n] such that does not contain any sunﬂowers with r petals. Then there exists a constantFD(r, α) such that

k (log log log(k ℓ))2 k ℓ D(r, α) (k ℓ)! − − . |F| ≤ ℓ − α log log(k ℓ) − Corollary 1.9 Let r > 2, k > 1 be ﬁxed integers. Let ℓ := k k . Let ⌈ − log k ⌉ be an ℓ-intersecting, k-uniform family of subsets of [n] such that does notF contain any sunﬂowers with r petals. Then there exists a constantF D(r) such that D(r)4k. |F| ≤ We present our proofs in Section 2. We give some concluding remarks in Section 3.

2 Proofs

We start our proof with an elementary fact.

Lemma 2.1 Let 0 r n be integers. Then ≤ ≤ n n 1 − r ≤ r +1 if and only if r2 + (1 3n)r + n2 2n 0. − − ≥

4 3n 1 √5(n+1) Corollary 2.2 Let 0 r n be integers. If 0 r − − , then ≤ ≤ ≤ ≤ 2 n n 1 − r ≤ r +1

We use the following easy Lemma in the proof of our main results.

Lemma 2.3 Let 0 ℓ k 1 be integers. Then ≤ ≤ − 2k ℓ √5 − 8 2(1+ 5 )k. ℓ +1 ≤ ·

Proof. First suppose that

√5 2√5 ℓ (1 )k (1 + ) . ≤ ⌈ − 5 − 5 ⌉ Then √ √ 2k ℓ 2k (1 5 )k (1 + 2 5 ) − − ⌈ − 5 − 5 ⌉ ℓ +1 ≤ (2 √5 )k (1 + 2√5 ) ≤ ⌈ − 5 − 5 ⌉ 5 2 5 5 2k (1 √ )k (1+ √ ) (1+ √ )k 2 −⌈ − 5 − 5 ⌉ 8 2 5 . ≤ ≤ · The ﬁrst inequality follows easily from Corollary 2.2. Namely if

√5 2√5 ℓ (1 )k (1 + ) , ≤ ⌈ − 5 − 5 ⌉ then 3 √5 1+ √5 ℓ +1 − (2k ℓ) ≤ ⌈ 2 − − 2 ⌉ and we can apply Corollary 2.2 with the choices r := ℓ + 1 and n := 2k ℓ. −

5 Secondly, suppose that

√5 2√5 ℓ> (1 )k (1 + ) . ⌈ − 5 − 5 ⌉ Then √5 2√5 √5 2k ℓ 2k (1 )k (1 + ) 2+(1+ )k , − ≤ − ⌈ − 5 − 5 ⌉ ≤ ⌈ 5 ⌉ hence 2k ℓ 2k (1 √5 )k (1 + 2√5 ) − − ⌈ − 5 − 5 ⌉ ℓ +1 ≤ ℓ +1 ≤

5 2 5 5 2k (1 √ )k (1+ √ ) (1+ √ )k 2 −⌈ − 5 − 5 ⌉ 8 2 5 . ≤ ≤ ·

The soul of the proof of our main result is the following Lemma.

Lemma 2.4 Let be an ℓ-intersecting, k-uniform family of subsets of [n] such that does notF contain any sunﬂowers with three petals. Suppose that F there exist F , F distinct subsets such that F F = ℓ. Let M := 1 2 ∈ F | 1 ∩ 2| F1 F2. Then ∪ F M >ℓ | ∩ | for each F . ∈F

Proof. Clearly F F F M for each F , hence ∩ 1 ⊆ ∩ ∈F F M ℓ | ∩ | ≥ for each F . ∈F We prove by an indirect argument. Suppose that there exists an F such that F M = ℓ. Clearly F = F and F = F . Let G := F F . Then∈F | ∩ | 6 1 6 2 1 ∩ 2 G = ℓ by assumption. It follows from F F F M and ℓ F F | | ∩ 1 ⊆ ∩ ≤| ∩ 1| ≤ F M = ℓ that F F1 = F M. Similarly F F2 = F M. Consequently |F ∩F =| F F . We∩ get that∩F F = F G =∩F F . Since∩ ℓ = F F = ∩ 1 ∩ 2 ∩ 2 ∩ ∩ 1 | ∩ 1|

6 F G G = ℓ and F G G, hence G = F G = F F2 = F F1, so | F,∩ F ,|≤| F is| a sunﬂower∩ with⊆ three petals, a contradiction.∩ ∩ ∩ { 1 2}

Proof of Theorem 1.7: We apply induction on L = s. If s = 1, then our result follows from Theorem 1.6. | | Suppose that Theorem 1.7 is true for s 1 and now we attack the case − L = s. | | If F F ′ = ℓ holds for each distinct F, F ′ , then is actually an | ∩ | 6 1 ∈ F F L′ := ℓ ,...,ℓ -intersecting system and the much stronger upper bound { 2 s} 5 2 (s 2) (1+ √ )k(s 2) (k k + 2)8 − 2 5 − . |F| ≤ − follows from the induction. Hence we can suppose that there exist F , F such that F F = ℓ . 1 2 ∈F | 1 ∩ 2| 1 Let M := F1 F2. Clearly is an ℓ1-intersecting family. It follows from Lemma 2.4 that∪ F F M >ℓ (1) | ∩ | 1 for each F . Clearly M =2k ℓ1. Let T be∈F a ﬁxed subset| of| M such− that T = ℓ + 1. Deﬁne the family | | 1 (T ) := F : T M F . F { ∈F ⊆ ∩ }

Let L′ := ℓ ,...,ℓ . Clearly L′ = s 1. Then (T ) is an L′-intersecting, { 2 s} | | − F k-uniform family, because is an L-intersecting family and T F for each F (T ). The following PropositionF follows easily from (1). ⊆ ∈F Proposition 2.5 = (T ). F F T M,[T =ℓ1+1 ⊆ | |

Let T be a ﬁxed, but arbitrary subset of M such that T = ℓ1 + 1. Consider the set system | |

(T ) := F T : F (T ) . G { \ ∈F }

7 Clearly (T ) = (T ) . Let L := ℓ ℓ 1,...,ℓ ℓ 1 . Here |G | |F | { 2 − 1 − s − 1 − } L = s 1. Since (T ) is an L′-intersecting, k-uniform family, thus (T ) is | | − F G an L-intersecting, (k ℓ1 1)-uniform family. It follows from the inductional hypothesis that − −

5 2 (s 2) (1+ √ )k(s 2) (T ) = (T ) (k k + 2)8 − 2 5 − . |F | |G | ≤ − Finally Proposition 2.5 implies that

5 2k ℓ1 2 (s 2) (1+ √ )k(s 2) (T ) − (k k + 2)8 − 2 5 − . |F| ≤ |F | ≤ ℓ1 +1 − T M,XT =ℓ1+1 ⊆ | | But 2k ℓ √5 − 8 2(1+ 5 )k ℓ +1 ≤ · by Lemma 2.3, hence

5 5 2 (s 2) (1+ √ )k(s 2) (1+ √ )k (k k + 2)8 − 2 5 − 8 2 5 = |F| ≤ − · · 5 2 (s 1) (1+ √ )k(s 1) =(k k + 2)8 − 2 5 − , − which was to be proved. Proof of Theorem 1.8: Let F0 be a ﬁxed subset. Clearly F F0 ℓ for each F , since is an ℓ-intersecting∈F family. | ∩ | ≥ ∈F F Let T be a ﬁxed, but arbitrary subset of F0 such that T = ℓ. Consider the set system | | (T ) := F : T F . F { ∈F ⊆ } It is easy to see the following Proposition.

Proposition 2.6 = (T ). F F T F[0, T =ℓ ⊆ | |

Deﬁne the set system

(T ) := F T : F (T ) . G { \ ∈F } 8 Obviously (T ) = (T ) and (T ) is a (k ℓ)-uniform family. It is easy|G to see| that|F (|T ) doesG not contain− any sunﬂowers with r petals, G because does not contain any sunﬂowers with r petals. Hence for each α F (log log log(k ℓ))2 k ℓ (T ) = (T ) D(r, α)(k ℓ)! − − |F | |G | ≤ − α log log(k ℓ) − by Theorem 1.2. Finally Proposition 2.5 implies that (T ) |F| ≤ |F | ≤ T FX0, T =ℓ ⊆ | | k (log log log(k ℓ))2 k ℓ D(r, α) (k ℓ)! − − . ≤ ℓ − α log log(k ℓ) −

Proof of Theorem 1.9: If ℓ := k k , then it is easy to verify that ⌈ − log k ⌉ k D(r) (k ℓ)! D(r)4k. ℓ − ≤ 3 Remarks

Deﬁne f(k,r,s) as the least number so that if is an arbitrary k-uniform, F L-intersecting family, where L = s, then > f(k,r,s) implies that contains a sunﬂower with r petals.| | In Theorem|F| 1.7 we proved the followingF recursion for f(k,r,s): 2k ℓ f(k, 3,s) max − f(k 1, 3,s 1). 0 ℓ k 1 ℓ ≤ ≤ ≤ − +1 − − Our upper bound in Theorem 1.7 was a clear consequence of this recursion. It would be very interesting to give a similar recursion for f(k,r,s) for r> 3. On the other hand, it is easy to prove the following Proposition from Theorem 1.3. Proposition 3.1 Let 1 s 0 positive constant such that≤ s/2 c log s f(k, 3,s) > 2 10 − ·

9 References

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