9,414 research outputs found

    Triangle-Intersecting Families of Graphs

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    A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.Comment: 43 page

    On symmetric intersecting families

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    We make some progress on a question of Babai from the 1970s, namely: for n,k∈Nn, k \in \mathbb{N} with k≀n/2k \le n/2, what is the largest possible cardinality s(n,k)s(n,k) of an intersecting family of kk-element subsets of {1,2,…,n}\{1,2,\ldots,n\} admitting a transitive group of automorphisms? We give upper and lower bounds for s(n,k)s(n,k), and show in particular that s(n,k)=o((nβˆ’1kβˆ’1))s(n,k) = o (\binom{n-1}{k-1}) as nβ†’βˆžn \to \infty if and only if k=n/2βˆ’Ο‰(n)(n/log⁑n)k = n/2 - \omega(n)(n/\log n) for some function Ο‰(β‹…)\omega(\cdot) that increases without bound, thereby determining the threshold at which `symmetric' intersecting families are negligibly small compared to the maximum-sized intersecting families. We also exhibit connections to some basic questions in group theory and additive number theory, and pose a number of problems.Comment: Minor change to the statement (and proof) of Theorem 1.4; the authors thank Nathan Keller and Omri Marcus for pointing out a mistake in the previous versio

    Minimum saturated families of sets

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    We call a family F\mathcal{F} of subsets of [n][n] ss-saturated if it contains no ss pairwise disjoint sets, and moreover no set can be added to F\mathcal{F} while preserving this property (here [n]={1,…,n}[n] = \{1,\ldots,n\}). More than 40 years ago, Erd\H{o}s and Kleitman conjectured that an ss-saturated family of subsets of [n][n] has size at least (1βˆ’2βˆ’(sβˆ’1))2n(1 - 2^{-(s-1)})2^n. It is easy to show that every ss-saturated family has size at least 12β‹…2n\frac{1}{2}\cdot 2^n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2+Ξ΅)2n(1/2 + \varepsilon)2^n, for some fixed Ξ΅>0\varepsilon > 0, seems difficult. In this note, we prove such a result, showing that every ss-saturated family of subsets of [n][n] has size at least (1βˆ’1/s)2n(1 - 1/s)2^n. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on ∣F1∣+…+∣Fs∣|\mathcal{F}_1| + \ldots + |\mathcal{F}_s| where F1,…,Fs\mathcal{F}_1, \ldots, \mathcal{F}_s are families of subsets of [n][n], such that there are no ss pairwise disjoint sets, one from each family Fi\mathcal{F}_i, and furthermore no set can be added to any of the families while preserving this property. We show that ∣F1∣+…+∣Fs∣β‰₯(sβˆ’1)β‹…2n|\mathcal{F}_1| + \ldots + |\mathcal{F}_s| \ge (s-1)\cdot 2^n, which is tight e.g.\ by taking F1\mathcal{F}_1 to be empty, and letting the remaining families be the families of all subsets of [n][n].Comment: 8 page

    Invariance principle on the slice

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    We prove an invariance principle for functions on a slice of the Boolean cube, which is the set of all vectors {0,1}^n with Hamming weight k. Our invariance principle shows that a low-degree, low-influence function has similar distributions on the slice, on the entire Boolean cube, and on Gaussian space. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. Our result imply a version of majority is stablest for functions on the slice, a version of Bourgain's tail bound, and a version of the Kindler-Safra theorem. As a corollary of the Kindler-Safra theorem, we prove a stability result of Wilson's theorem for t-intersecting families of sets, improving on a result of Friedgut.Comment: 36 page
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