论文标题
$（k+2，k）$的极限 - brown，erdős和sós的问题均存在所有$ k \ geq 2 $
The limit in the $(k+2, k)$-Problem of Brown, Erdős and Sós exists for all $k\geq 2$
论文作者
论文摘要
令$ f^{（r）}（n; s，k）$是〜$ n $ vertices上不包含带有$ k $〜边缘的子图的$ r $ - 均匀超图的最大边数。 1973年，Brown，Erdős和Sós猜想$$ \ lim_ {n \ to \ infty} n^{ - 2} f^{（3）}（n; k+2，k）$ k+ge 2 $。他们以$ k = 2 $证明了这一点。 2019年，格洛克（Glock）以$ k = 3 $的形式证明了这一点，并确定了限制。最近，Glock，Joos，Kim，Kühn，Lichev和Pikhurko证明了这一点，以$ K = 4 $，并确定了极限。我们将他们的工作与新的减少结合在一起，以证明所有正整数$ k \ ge 2 $的限制确实存在。
Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniform hypergraph on~$n$ vertices not containing a subgraph with $k$~edges and at most $s$~vertices. In 1973, Brown, Erdős and Sós conjectured that the limit $$\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 $k\ge 2$.
