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The profile polytope of non-trivial intersecting families

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arxiv 2109.05615 v2 pith:Y2FNZZJL submitted 2021-09-12 math.CO

Dániel Gerbner This is my paper · ORCID
classification math.CO
keywords profileelementfamiliesintersectingmathcalnon-trivialclassdenotes
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abstract

The profile vector of a family $\mathcal{F}$ of subsets of an $n$-element set is $(f_0,f_1, \ldots, f_n)$ where $f_i$ denotes the number of the $i$-element members of $\mathcal{F}$. In this paper we determine the extreme points of the set of profile vectors for the class of non-trivial intersecting families.

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  1. A gap theorem for non-trivial maximal intersecting families and an exact weighted asymptotic

    math.CO 2026-07 accept novelty 7.0 of 10

    Among non-trivial maximal intersecting families, every family other than the n one-flip stars has weight exponent at least 2^{n-2}-4 below the maximum, yielding the exact prefactor R(n)=(3/4+o(1)) n 2^{3^{n-1}-2^{n-1}+2}.

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