On the $(6,4)$-problem of Brown, Erd\H{o}s and S\'os

TL;DR

This paper determines the asymptotic maximum edge count in certain hypergraphs avoiding specific subgraphs, settling a longstanding conjecture for the case k=4 and generalizing to other parameters.

Contribution

It proves the existence and exact value of limits for maximum edges in r-uniform hypergraphs avoiding particular subgraphs, resolving open cases for k=4 and general parameters.

Findings

Established the limit for f^{(3)}(n;6,4) as (7/36+o(1))n^2.

Computed the limit for f^{(r)}(n;k(r-t)+t,k) for all k in {3,4}, r≥3, t in [2,r-1].

Settled a problem posed by Shangguan and Tamo.

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 $k$ and confirmed it for $k=2$. Recently, Glock showed this for $k=3$. We settle the next open case, $k=4$, by showing that $f^{(3)}(n;6,4)=\left(\frac{7}{36}+o(1)\right)n^2$ as $n\to\infty$. More generally, for all $k\in \{3,4\}$, $r\ge 3$ and $t\in [2,r-1]$, we compute the value of the limit $\lim_{n\to \infty} n^{-t}f^{(r)}(n;k(r-t)+t,k)$, which settles a problem of Shangguan and Tamo.