On the $(k+2,k)$-problem of Brown, Erd\H{o}s and S\'{o}s for even integers $k$

TL;DR

This paper determines the exact asymptotic maximum edge density for certain uniform hypergraphs with constraints on the number of vertices spanned by edges, extending previous results to all even integers k and specific uniformities r.

Contribution

It provides the exact limit value of the maximum edge density for even k ≥ 4 and r ≥ 2 + sqrt(3/2 * k - 4), generalizing prior work and confirming a conjectured formula.

Findings

Exact limit value is 1/(r^2 - r) for specified parameters.

Extends previous results to all even k ≥ 4 and certain r.

Confirms the conjectured asymptotic density formula.

Abstract

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $r$-graph on $n$ vertices in which every $k$ edges span more than $s$ vertices. Brown, Erd\H{o}s and S\'{o}s in 1973 conjectured that for every $k\geq 2$, the limit $\lim_{n\to\infty} n^{-2} f^{(3)}(n;k+2,k)$ exists and verified the conjecture for $k=2$ by showing that $\lim_{n\to\infty} n^{-2} f^{(3)}(n;4,2)=\frac{1}{6}$. Delcourt and Postle, building on the work of Glock, Joos, Kim, K\"uhn, Lichev and Pikhurko, proved that for every $k\geq 2$, the limit $\lim_{n\to\infty} n^{-2} f^{(3)}(n;k+2,k)$ exists, thereby solving this conjecture. Their approach was later generalised by Shangguan to every uniformity $r\geq 4$: the limit $\lim_{n\to\infty} n^{-2} f^{(r)}(n; rk-2k+2,k)$ exists for all $r\geq 3$ and $k\geq 2$. However, its exact value was not determined. When $k\in\{2,3,\ldots,7\}$, the exact values of $\lim_{n\to\infty} n^{-2} f^{(r)}(n; rk-2k+2,k)$ were determined by Glock, Joos, Kim, K\"{u}hn, Lichev, Pikhurko, R\"{o}dl and Sun. Very recently, the limit for $k=8$ and $r\geq 4$ was determined by Pikhurko and Sun. For a general even integer $k$, Letzter and Sgueglia obtained the exact values of $\lim_{n\to\infty} n^{-2} f^{(r)}(n;rk-2k+2,k)$ for every even integer $k$ and uniformity $r\geq 2+\sqrt{2}\,k^{3/2}$. In this paper, we determine the exact value of $\lim_{n\to\infty} n^{-2} f^{(r)}(n;rk-2k+2,k)$ for every even integer $k\geq 4$ and $r\geq 2+\sqrt{\frac{3}{2}k-4}$, and show that it is $\frac{1}{r^2-r}.$