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arXiv · 2210.01105

The limit in the $(k+2, k)$-Problem of Brown, Erdős and Sós exists for all $k\geq 2$

Michelle Delcourt·Luke Postle
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Abstract

Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniform hypergraph on~$n$ vertices not containing a subgraph with $k$~edges and at most $s$~vertices. In 1973, Brown, Erdős and Sós conjectured that the limit $$\lim_{n\to \infty} n^{-2} f^{(3)}(n;k+2,k)$$ exists for all positive integers $k\ge 2$. They proved this for $k=2$. In 2019, Glock proved this for $k=3$ and determined the limit. Quite recently, Glock, Joos, Kim, Kühn, Lichev and Pikhurko proved this for $k=4$ and determined the limit; we combine their work with a new reduction to fully resolve the conjecture by proving that indeed the limit exists for all positive integers $k\ge 2$.

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Michelle Delcourt, Luke Postle. 2023-09-14. The limit in the $(k+2, k)$-Problem of Brown, Erdős and Sós exists for all $k\geq 2$. https://arxiv.org/abs/2210.01105

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