85,835 research outputs found

    On Mubayi's Conjecture and conditionally intersecting sets

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    Mubayi's Conjecture states that if F\mathcal{F} is a family of kk-sized subsets of [n]={1,…,n}[n] = \{1,\ldots,n\} which, for kβ‰₯dβ‰₯2k \geq d \geq 2, satisfies A1βˆ©β‹―βˆ©Adβ‰ βˆ…A_1 \cap\cdots\cap A_d \neq \emptyset whenever ∣A1βˆͺβ‹―βˆͺAdβˆ£β‰€2k|A_1 \cup\cdots\cup A_d| \leq 2k for all distinct sets A1,…,Ad∈FA_1,\ldots,A_d \in\mathcal{F}, then ∣Fβˆ£β‰€(nβˆ’1kβˆ’1)|\mathcal{F}|\leq \binom{n-1}{k-1}, with equality occurring only if F\mathcal{F} is the family of all kk-sized subsets containing some fixed element. This paper proves that Mubayi's Conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayi's Conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between (i,j)(i,j)-unstable families and (j,i)(j,i)-unstable families. Generalising previous intersecting conditions, we introduce the (d,s,t)(d,s,t)-conditionally intersecting condition for families of sets and prove general results thereon. We conjecture on the size and extremal structures of families F∈([n]k)\mathcal{F}\in\binom{[n]}{k} that are (d,2k)(d,2k)-conditionally intersecting but which are not intersecting, and prove results related to this conjecture. We prove fundamental theorems on two (d,s)(d,s)-conditionally intersecting families that generalise previous intersecting families, and we pose an extension of a previous conjecture by Frankl and F\"uredi on (3,2kβˆ’1)(3,2k-1)-conditionally intersecting families. Finally, we generalise a classical result by Erd\H{o}s, Ko and Rado by proving tight upper bounds on the size of (2,s)(2,s)-conditionally intersecting families FβŠ†2[n]\mathcal{F}\subseteq 2^{[n]} and by characterising the families that attain these bounds. We extend this theorem for certain parametres as well as for sufficiently large families with respect to (2,s)(2,s)-conditionally intersecting families FβŠ†2[n]\mathcal{F}\subseteq 2^{[n]} whose members have at most a fixed number uu members

    Intersecting families of discrete structures are typically trivial

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    The study of intersecting structures is central to extremal combinatorics. A family of permutations FβŠ‚Sn\mathcal{F} \subset S_n is \emph{tt-intersecting} if any two permutations in F\mathcal{F} agree on some tt indices, and is \emph{trivial} if all permutations in F\mathcal{F} agree on the same tt indices. A kk-uniform hypergraph is \emph{tt-intersecting} if any two of its edges have tt vertices in common, and \emph{trivial} if all its edges share the same tt vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for nn sufficiently large with respect to tt, the largest tt-intersecting families in SnS_n are the trivial ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest tt-intersecting kk-uniform hypergraphs are also trivial when nn is large. We determine the \emph{typical} structure of tt-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollob\'as set-pairs inequality to bound the number of maximal intersecting families, which can then be combined with known stability theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira result. Update 2: corrected statement of the unpublished Hamm--Kahn result, and slightly modified notation in Theorem 1.6 Update 3: new title, updated citations, and some minor correction

    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
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