A Proof and Generalizations of the Erdős-Ko-Rado Theorem Using the Method of Linearly Independent Polynomials

Füredi, Zoltán; Hwang, Kyung-Won; Weichsel, Paul M.

doi:10.1007/3-540-33700-8_13

Cited by 13 publications

(8 citation statements)

References 19 publications

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“…There are several proofs and extensions of Theorem 1.1 in the literature [3,4,[6][7][8]10,11,15]. In particular, Deza and Frankl [10] extended Theorem 1.1 for permutations.…”

Section: Introductionmentioning

confidence: 99%

2-intersecting permutations

Meagher,

Razafimahatratra

2020

Preprint

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In this paper we consider the Erdős-Ko-Rado property for both 2-pointwise and 2-setwise intersecting permutations. Two permutations σ, τ ∈ Sym(n) are t-setwise intersecting if there exists a t-subset S of {1, 2, . . . , n} such that S σ = S τ . If for each s ∈ S, s σ = s τ , then we say σ and τ are t-pointwise intersecting. We say that Sym(n) has the t-setwise (resp. t-pointwise) intersecting property if for any family F of t-setwiseEllis (["Setwise intersecting families of permutations". Journal of Combinatorial Theory, Series A, 119(4):825849, 2012.]), proved that for n sufficiently large relative to t, Sym(n) has the tsetwise intersecting property. Ellis also conjuctured that this result holds for all n ≥ t. Ellis, Friedgut and Pilpel [Ellis, David, Ehud Friedgut, and Haran Pilpel. "Intersecting families of permutations." Journal of the American Mathematical Society 24(3):649-682, 2011.] also proved that for n sufficiently large relative to t, Sym(n) has the t-pointwise intersecting property. It is also conjectured that Sym(n) has the t-pointwise intersecting propoperty for n ≥ 2t + 1. In this work, we prove these two conjectures for Sym(n) when t = 2.

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“…There are several proofs and extensions of Theorem 1.1 in the literature [3,4,[6][7][8]10,11,15]. In particular, Deza and Frankl [10] extended Theorem 1.1 for permutations.…”

Section: Introductionmentioning

confidence: 99%

2-intersecting permutations

Meagher,

Razafimahatratra

2020

Preprint

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“…The method of linearly independent polynomials has also been used to prove many intersection theorems about set families by Blokhuis [11], Chen and Liu [12], Furedi, Hwang, and Weichsel [13], Liu and Yang [14], Qian and Ray-Chaudhuri [15], Ramanan [16], Snevily [17,18], Wang, Wei, and Ge [19], and others.…”

Section: Theorem 8 (Frankl and Wilsonmentioning

confidence: 99%

An Erdős-Ko-Rado Type Theorem via the Polynomial Method

2020

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A family F is an intersecting family if any two members have a nonempty intersection. Erdős, Ko, and Rado showed that | F | ≤ n − 1 k − 1 holds for a k-uniform intersecting family F of subsets of [ n ] . The Erdős-Ko-Rado theorem for non-uniform intersecting families of subsets of [ n ] of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that | F | ≤ n − 1 k − 1 + n − 1 k − 2 + ⋯ + n − 1 0 holds for non-uniform intersecting families of subsets of [ n ] of size at most k. In this paper, we prove that the same upper bound of the Erdős-Ko-Rado Theorem for k-uniform intersecting families of subsets of [ n ] holds also in the non-uniform family of subsets of [ n ] of size at least k and at most n − k with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials.

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“…which implies (2). For arbitrary prescribed partitions (3) is proved using the polynomials described in [1].…”

Section: Upper Boundsmentioning

confidence: 99%

“…Furthermore, for all n ≥ 2k − 1 and k ≥ 2 there exist partition critical k-uniform hypergraphs of size n k−1 . The proof of the upper bound (2) is based on the polynomial method outlined in [2].…”

Section: Theorem 15 Let E ⊆ [N]mentioning

confidence: 99%