On the $(k+2,k)$-problem of Brown, Erdős and Sós for even integers $k$

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, Erdős and Sós in 1973 conjectured that for every $k\ge 2$, 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, Kühn, Lichev and Pikhurko, proved that for every $k\ge 2$, 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 $r\ge 4$: the limit ${\lim }_{n\to \infty }{n}^{-2}{f}^{(r)}(n;rk-2k+2,k)$ exists for all $r\ge 3$ and $k\ge 2$. However, its exact value was not determined. When $k\in \{2,3,\dots ,7\}$, the exact values of ${\lim }_{n\to \infty }{n}^{-2}{f}^{(r)}(n;rk-2k+2,k)$ were determined by Glock, Joos, Kim, Kühn, Lichev, Pikhurko, Rödl and Sun. Very recently, the limit for $k=8$ and $r\ge 4$ 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 $r\ge 2+\sqrt{2}{k}^{3/2}$. 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 $k\ge 4$ and $r\ge 2+\sqrt{\frac{3}{2}k-4}$, and show that it is $\frac{1}{r^2-r}.$

13 pages, 1 figure