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arXiv · 2002.01498

Andr\'asfai and Vega graphs in Ramsey-Tur\'an theory

Tomasz Łuczak·Joanna Polcyn·Christian Reiher
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Abstract

Given positive integers $n\ge s$, we let ${\mathrm{ex}}(n,s)$ denote the maximum number of edges in a triangle-free graph $G$ on $n$ vertices with $\alpha(G)\le s$. In the early sixties Andr\'{a}sfai conjectured that for $n/3<s<n/2$ the function ${\mathrm{ex}}(n, s)$ is piecewise quadratic with critical values at $s/n={k}/({3k-1})$. We confirm that this is indeed the case whenever $s/n$ is slightly larger than a critical value, thus determining ${\mathrm{ex}}(n,s)$ for all $n$ and $s$ such that $s/n\in [{k}/({3k-1}), {k}/({3k-1})+\gamma_k]$, where $\gamma_k=\Theta(k^{-6})$.

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Tomasz Łuczak, Joanna Polcyn, Christian Reiher. 2020-02-04. Andr\'asfai and Vega graphs in Ramsey-Tur\'an theory. https://doi.org/10.1002/jgt.22682

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