The limit in the $(k+2, k)$-Problem of Brown, Erd\H{o}s and S\'os exists for all $k\geq 2$

TL;DR

This paper proves that the limit of the maximum number of edges in certain hypergraphs, avoiding specific subgraphs, exists for all positive integers k, resolving a long-standing conjecture in combinatorics.

Contribution

The authors establish the existence of the limit for all k ≥ 2, extending previous results that confirmed it for small values of k, thus fully resolving the conjecture.

Findings

The limit exists for all k ≥ 2.

The proof combines recent results with a new reduction technique.

Resolved a 50-year-old conjecture in hypergraph theory.

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, Erd\H{o}s and S\'os 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, K\"{u}hn, 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 $k\ge 2$.