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The edge-statistics conjecture for $\ell \ll k^{6/5}$

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arxiv 1809.02576 v3 pith:TCNRX5DK submitted 2018-09-07 math.CO math.PR

Anders Martinsson , Frank Mousset , Andreas Noever , Miloš Trujić This is my paper · ORCID
classification math.COmath.PR
keywords conjecturealonedge-statisticsedgesenougheveryexactlyfraction
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abstract

Let $k$ and $\ell$ be positive integers. We prove that if $1 \leq \ell \leq o_k(k^{6/5})$, then in every large enough graph $G$, the fraction of $k$-vertex subsets that induce exactly $\ell$ edges is at most $1/e + o_k(1)$. Together with a recent result of Kwan, Sudakov, and Tran, this settles a conjecture of Alon, Hefetz, Krivelevich, and Tyomkyn.

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  1. An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

    math.CO 2019-09 accept novelty 8.0 of 10

    If a quadratic Bernoulli polynomial has a point probability much larger than 1/n, it is close to a quadratic form of low rank; a consequence bounds edge-count point probabilities in Ramsey graphs by n^{-1+o(1)}.

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