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• W2154099473 abstract "A d-dimensional simplex is a collection of d+1 sets with empty intersection, every d of which have nonempty intersection. A k-uniform d-cluster is a collection of d+1 sets of size k with empty intersection and union of size at most 2k. We prove the following result which simultaneously addresses an old conjecture of Chvátal [6] and a recent conjecture of the second author [28]. For d≥2 and ζ >0 there is a number T such that the following holds for sufficiently large n. Let G be a k-uniform set system on [n] ={1,…,n} with ζ n<k <n/2−T, and suppose either that G contains no d-dimensional simplex or that G contains no d-cluster. Then |G|≤ % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWaaeWaaeaafaqabeGabaaabaGaemOBa4MaeyOe % I0IaeGymaedabaGaem4AaSMaeyOeI0IaeGymaedaaaGaayjkaiaawM % caaaaa!43EB! $$ left( {begin{array}{*{20}c} {n - 1} {k - 1} end{array} } right) $$ with equality only for the family of all k-sets containing a specific element. In the non-uniform setting we obtain the following exact result that generalises a question of Erdős and a result of Milner, who proved the case d=2. Suppose d≥2 and G is a set system on [n] that does not contain a d-dimensional simplex, with n sufficiently large. Then |G|≤2n−1+Σ =0 −1 % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWaaeWaaeaafaqabeGabaaabaGaemOBa4MaeyOe % I0IaeGymaedabaGaemyAaKgaaaGaayjkaiaawMcaaaaa!420A! $$ left( {begin{array}{*{20}c} {n - 1} i end{array} } right) $$ , with equality only for the family consisting of all sets that either contain some specific element or have size at most d−1. Each of these results is proved via the corresponding stability result, which gives structural information on any G whose size is close to maximum. These in turn rely on a stability theorem that we obtain using an earlier result of Frankl." @default.
• W2154099473 created "2016-06-24" @default.
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• W2154099473 date "2010-03-01" @default.
• W2154099473 modified "2023-09-27" @default.
• W2154099473 title "Set systems without a simplex or a cluster" @default.
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• W2154099473 doi "https://doi.org/10.1007/s00493-010-2401-x" @default.
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