论文标题
在$（6,4）$ - Brown，Erdős和Sós的问题上
On the $(6,4)$-problem of Brown, Erdős and Sós
论文作者
论文摘要
令$ f^{（r）}（n; s，k）$为$ r $均匀的超图的最大边数，上面$ n $ n $顶点不包含带有$ k $ edges的子图的边缘，最多最多$ s $ s $ vertices。 1973年，Brown，Erdős和Sós推测$ \ lim_ {n \ to \ infty} n^{ - 2} f^{（3）}（n; k+2，k）$ k $都存在，并确认了$ k = 2 $。最近，格洛克（Glock）以$ k = 3 $表示了这一点。我们通过表明$ f^{（3）}（n; 6,4）= \ left（\ frac {7} {36} {36}+o（1）\ right）n^2 $ as $ n \ to \ infty $来解决下一个开放式案例，$ k = 4 $，$ k = 4 $。更笼统地，对于所有$ k \ in \ {3,4 \} $，$ r \ ge 3 $和$ t \ in [2，r-1] $，我们计算了极限$ \ lim_ {n \ to \ lim_ {n \ to \ infty} n^{ - t} n^{ - t} n^{ - t} f^{（r）（r）}（n;
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 $k$ 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.
