aa r X i v : . [ m a t h . C O ] D ec A generalization of the Erd˝os-Ko-RadoTheorem
G´abor Heged˝us ´Obuda University
August 20, 2018
Abstract
Our main result is a new upper bound for the size of k -uniform, L -intersecting families of sets, where L contains only positive inte-gers. We characterize extremal families in this setting. Our proof isbased on the Ray-Chaudhuri–Wilson Theorem. As an application, wegive a new proof for the Erd˝os-Ko-Rado Theorem, improve Fisher’sinequality in the uniform case and give an uniform version of theFrankl-F¨uredi conjecture . Keywords. L -intersecting families, Erd˝os-Ko-Rado Theorem, ex-tremal set theory First we introduce some notations.Let [ n ] stand for the set { , , . . . , n } . We denote the family of all subsetsof [ n ] by 2 [ n ] .For an integer 0 ≤ k ≤ n we denote by (cid:0) [ n ] k (cid:1) the family of all k elementsubsets of [ n ].We call a family F of subsets of [ n ] k -uniform , if | F | = k for each F ∈ F .Bose proved the following result in [1]. Theorem 1.1
Let λ > be a positive integer. Let F = { F , . . . , F m } be a k -uniform family of subsets of [ n ] such that | F i ∩ F j | = λ for each ≤ i, j ≤ m, i = j . Then m ≤ n . Theorem 1.2
Let λ > be a positive integer. Let F = { F , . . . , F m } be afamily of subsets of [ n ] such that | F i ∩ F j | = λ for each ≤ i, j ≤ m, i = j .Then m ≤ n . Frankl and F¨uredi conjectured in [7] and Ramanan proved in [10] thefollowing statement.
Theorem 1.3
Let F = { F , . . . , F m } be a family of subsets of [ n ] such that ≤ | F i ∩ F j | ≤ s for each ≤ i, j ≤ m, i = j . Then m ≤ s X i =0 (cid:18) n − i (cid:19) . Later Snevily conjectured the following statement in his doctoral disser-tation (see [13]). Finally he proved this result in [12].
Theorem 1.4
Let F = { F , . . . , F m } be a family of subsets of [ n ] . Let L = { ℓ , . . . , ℓ s } be a collection of s positive integer. If | F i ∩ F j | ∈ L for each ≤ i, j ≤ m, i = j , then m ≤ s X i =0 (cid:18) n − i (cid:19) . A family F is t -intersecting , if | F ∩ F ′ | ≥ t whenever F, F ′ ∈ F . Specially, F is an intersecting family, if F ∩ F ′ = ∅ whenever F, F ′ ∈ F .Erd˝os, Ko and Rado proved the following well-known result in [6]: Theorem 1.5
Let n, k, t be integers with < t < k < n . Suppose F is a t -intersecting, k -uniform family of subsets of [ n ] . Then for n > n ( k, t ) , |F | ≤ (cid:18) n − tk − t (cid:19) . Further, |F | = (cid:0) n − tk − t (cid:1) if and only if for some T ∈ (cid:0) [ n ] t (cid:1) we have F = { F ∈ (cid:18) [ n ] k (cid:19) : T ⊆ F } . L be a set of nonnegative integers. A family F is L -intersecting , if | E ∩ F | ∈ L for every pair E, F of distinct members of F . The following theoremgives a remarkable upper bound for the size of a k -uniform L -intersectingfamily. Theorem 1.6 (Ray-Chaudhuri–Wilson) Let < s ≤ k ≤ n be positiveintegers. Let L be a set of s nonnegative integers and F = { F , . . . , F m } an L -intersecting, k -uniform family of subsets of [ n ] . Then m ≤ (cid:18) ns (cid:19) . Erd˝os, Deza and Frankl improved Theorem 1.6 in [5]. They used thetheory of ∆-systems in their proof.
Theorem 1.7
Let < s ≤ k ≤ n be positive integers. Let L be a set of s nonnegative integers and F = { F , . . . , F m } an L -intersecting, k -uniformfamily of subsets of [ n ] . Then for n > n ( k, L ) m ≤ s Y i =1 n − ℓ i k − ℓ i . Barg and Musin gave an improved version of Theorem 1.6 in [2].
Theorem 1.8
Let L be a set of s nonnegative integers and F = { F , . . . , F m } an L -intersecting, k -uniform family of subsets of [ n ] . Suppose that s ( k − ( s − k − n/ n − s − ≤ s X i =1 ℓ i . Then m ≤ (cid:18) ns (cid:19) − (cid:18) ns − (cid:19) n − s + 3 n − s + 2 . First we prove a special case of our main result.
Proposition 1.9
Let < s ≤ k ≤ n be positive integers. Let L = { ℓ , . . . , ℓ s } be a set of s positive integers such that < ℓ < . . . < ℓ s . Suppose that n ≥ (cid:0) k ℓ +1 (cid:1) s + ℓ . Let F = { F , . . . , F m } be an L -intersecting, k -uniformfamily of subsets of [ n ] . Suppose that T F ∈ F F = ∅ . Then m ≤ (cid:18) n − ℓ s (cid:19) .
3e state now our main results.
Theorem 1.10
Let < s ≤ k ≤ n be positive integers. Let L = { ℓ , . . . , ℓ s } be a set of s positive integers such that < ℓ < . . . < ℓ s . Suppose that n ≥ (cid:0) k ℓ +1 (cid:1) s + ℓ . Let G = { G , . . . , G m } be an L -intersecting, k -uniformfamily of subsets of [ n ] . Then m ≤ (cid:18) n − ℓ s (cid:19) . In the proof of Theorem 1.10 we combine simple combinatorial argumentswith the Ray-Chaudhuri–Wilson Theorem 1.6. Our proof was inspired by theproof of Proposition 8.8 in [8].In the following we characterize the extremal families appearing in The-orem 1.10.
Corollary 1.11
Let < s ≤ k ≤ n be positive integers. Let L = { ℓ , . . . , ℓ s } be a set of s positive integers such that < ℓ < . . . < ℓ s . Suppose that n > (cid:0) k ℓ +1 (cid:1) s + ℓ . Let G = { G , . . . , G m } be an L -intersecting, k -uniformfamily of subsets of [ n ] . Suppose that |G| = (cid:18) n − ℓ s (cid:19) . Then there exists a T ∈ (cid:0) [ n ] ℓ (cid:1) subset such that T ⊆ G for each G ∈ G . We give here some immediate consequences of Theorem 1.10. First wedescribe an uniform version of Theorem 1.3.
Corollary 1.12
Let < s < k ≥ n be positive integers. Let L = { , , . . . , s } .Suppose that n > (cid:0) k (cid:1) s . Let F = { F , . . . , F m } be an L -intersecting, k -uniform family of subsets of [ n ] . Then m ≤ (cid:18) n − s (cid:19) . Further if n > (cid:0) k (cid:1) s + 1 and |F | = (cid:18) n − s (cid:19) , then T F ∈ F F = ∅ . Corollary 1.13
Let λ > be a positive integer. Suppose that n ≥ (cid:0) k λ +1 (cid:1) + λ .Let F = { F , . . . , F m } be a k -uniform family of subsets of [ n ] such that | F i ∩ F j | = λ for each ≤ i, j ≤ m, i = j . Then m ≤ n − λ. Further if n > (cid:0) k λ +1 (cid:1) + λ and |F | = n − λ, then there exists a T ∈ (cid:0) [ n ] λ (cid:1) subset such that T ⊆ F for each F ∈ F . The following special case of Theorem 1.5 follows immediately from Theorem1.10.
Corollary 1.14
Let n, k, t be integers with < t < k < n . Suppose that n ≥ ( k − t ) (cid:0) k t +1 (cid:1) + t . Let F = { F , . . . , F m } be a t -intersecting, k -uniformfamily of subsets of [ n ] . Then m ≤ (cid:18) n − tk − t (cid:19) . Proof.
Let L := { t, t + 1 , . . . , k − } and apply Theorem 1.10.Similarly Corollary 1.11 implies the following result. Corollary 1.15
Let < k ≤ n be integers such that n > ( k − t ) (cid:0) k t +1 (cid:1) + t .Let F be a t -intersecting, k -uniform family of subsets of [ n ] . Suppose that |F | = (cid:18) n − tk − t (cid:19) . Then there exists a T ∈ (cid:0) [ n ] t (cid:1) subset such that T ⊆ F for each F ∈ F . Proof
The following Lemma is a well-known Helly-type result (see e.g. [3]).
Lemma 2.1
If each family of at most k + 1 members of a k -uniform setsystem intersect, then all members intersect. In our proof we use the following Lemma.
Lemma 2.2
Let ℓ be a positive integer. Let H be a family of subsets of [ n ] . Suppose that T H ∈ H H = ∅ . Let F ⊆ [ n ] , F / ∈ H be a subset such that | F ∩ H | ≥ ℓ for each H ∈ H . Let Q := S H ∈ H H . Then | Q ∩ F | ≥ ℓ + 1 . Proof.
Since | F ∩ H | ≥ ℓ for each H ∈ H , thus | Q ∩ F | ≥ ℓ . Indirectly,suppose that | Q ∩ F | = ℓ . Let U := Q ∩ F . Then U = Q ∩ F = ( [ H ∈ H H ) ∩ F = [ H ∈ H ( H ∩ F ) . Hence H ∩ F ⊆ U for each H ∈ H . Since | U | = ℓ and | H ∩ F | ≥ ℓ for each H ∈ H , thus U = H ∩ F for each H ∈ H . Hence U ⊆ T H ∈ H H , which is acontradiction with T H ∈ H H = ∅ . Lemma 2.3
Let H be a family of subsets of [ n ] . Suppose that t := |H| ≥ and H is a k -uniform, intersecting family. Then | [ H ∈ H H | ≤ k + ( t − k − . (1)6 roof. We use induction on t . The inequality (1) is trivially true for t = 2.Let t ≥
3. Suppose that the inequality (1) is true for t −
1. Let H bean arbitrary k -uniform intersecting family such that |H| = t . Let G ⊆ H be a fixed subset of H such that |G| = t −
1. Clearly G is intersecting and k -uniform. It follows from the induction hypothesis that | [ G ∈ G G | ≤ k + ( t − k − . Let { S } = H \ G . Then [ H ∈ H H = ( [ G ∈ G G ) ∪ S, thus | [ H ∈ H H | = | [ G ∈ G G | + | S |−| ( [ G ∈ G G ) ∩ S | ≤ k +( t − k − k − k +( t − k − . Proof of Proposition 1.9:
Consider the special case when T F ∈ F F = ∅ . By Lemma 2.1 there exists a G ⊆ F subset such that T G ∈ G G = ∅ and |G| = k + 1. Let M := [ G ∈ G G. It follows from Lemma 2.3 that | M | ≤ k + k ( k −
1) = k . On the other handit is easy to see that | M ∩ F | ≥ ℓ + 1 for each F ∈ F by Lemma 2.2.Let T be a fixed subset of M such that | T | = ℓ + 1. Define F ( T ) := { F ∈ F : T ⊆ M ∩ F } . Let L ′ := { ℓ , . . . , ℓ s } . Clearly | L ′ | = s −
1. Then F ( T ) is an L ′ -intersecting, k -uniform family, because F is an L -intersecting family and | M ∩ F | ≥ ℓ + 1for each F ∈ F . 7 roposition 2.4 F = [ T ⊆ M, | T | = ℓ +1 F ( T ) . Proof.
Let M := S T ⊆ M, | T | = ℓ +1 F ( T ). Clearly M ⊆ F . We prove that
F ⊆ M .Let F ∈ F be an arbitrary subset. Firstly, if F ∈ G , then F ∩ M = F ,because M = S G ∈ G G . Let T be a fixed subset of F such that | T | = ℓ + 1.Then F ∈ F ( T ). Secondly, suppose that F / ∈ G . Then | F ∩ M | ≥ ℓ + 1 byLemma 2.2. Let T be a fixed subset of F ∩ M such that | T | = ℓ + 1. Then F ∈ F ( T ) again.Let T be a fixed, but arbitrary subset of M such that | T | = ℓ + 1.Consider the set system G ( T ) := { F \ T : F ∈ F ( T ) } . Clearly |G ( T ) | = |F ( T ) | . Let L := { ℓ − ℓ − , . . . , ℓ s − ℓ − } . Here | L | = s −
1. Since F ( T ) is an L ′ -intersecting, k -uniform family, thus G ( T )is an L -intersecting, ( k − ℓ − G ⊆ [ n ] \ T for each G ∈ G ( T ). Hence it follows from Theorem 1.6 that |F ( T ) | = |G ( T ) | ≤ (cid:18) n − ℓ − s − (cid:19) . Finally Proposition 2.4 implies that |F | ≤ X T ⊆ M, | T | = ℓ +1 |F ( T ) | ≤ (cid:18) k ℓ + 1 (cid:19)(cid:18) n − ℓ − s − (cid:19) , but (cid:18) n − ℓ − s − (cid:19) = sn − ℓ (cid:18) n − ℓ s (cid:19) , hence |F | ≤ (cid:18) k ℓ + 1 (cid:19) sn − ℓ (cid:18) n − ℓ s (cid:19) ≤ (cid:18) n − ℓ s (cid:19) because n ≥ (cid:0) k ℓ +1 (cid:1) s + ℓ . 8 roof of Theorem 1.10: First we handle the case when | T G ∈ G G | ≥ ℓ . Let T be a fixed subset of T G ∈ G G such that | T | = ℓ . Consider the set system K := { G \ T : G ∈ G} . Obviously |G| = |K| . Let L ′ := { , ℓ − ℓ , . . . , ℓ s − ℓ } . Then clearly K isa ( k − ℓ )-uniform L ′ -intersecting set system of subsets of [ n ] \ T . It followsimmediately from Ray-Chaudhuri–Wilson Theorem 1.6 that |G| = |K| ≤ (cid:18) n − ℓ s (cid:19) . Now suppose that | T G ∈ G G | = t , where 0 < t < ℓ . Let T be a fixed subsetof T G ∈ G G such that | T | = t . Then consider the set system F := { G \ T : G ∈ G} . Clearly |F | = |G| . Let L ′ := { ℓ − t, ℓ − t, . . . , ℓ s − t } . Then clearly F isa ( k − t )-uniform L ′ -intersecting set system of subsets of [ n ] \ T . It followsfrom Proposition 1.9 that |G| = |F | ≤ (cid:18) n − t − ( ℓ − t ) s (cid:19) = (cid:18) n − ℓ s (cid:19) . Finally suppose that T G ∈ G G = ∅ . Then Proposition 1.9 gives us immedi-ately that |G| ≤ (cid:18) n − ℓ s (cid:19) . Proof of Corollary 1.11:
It follows from the proof of Theorem 1.10 that if |F | = (cid:0) n − ℓ s (cid:1) and n > (cid:0) k ℓ +1 (cid:1) s + ℓ , then | T F ∈ F F | ≥ ℓ . Thus there exists a T ∈ (cid:0) [ n ] ℓ (cid:1) such that T ⊆ F for each F ∈ F . 9 Remarks
Let q ≥ P G (2 , q ) denote the finiteprojective plane over the Galois field GF ( q ). Denote by L the set of all linesof P G (2 , q ). Let k := q + 1. Then L can be considered as a k -uniform familyof subsets of the base set [ k − k + 1]. Clearly |L| = k .This example motivates our next conjecture. Conjecture 1
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