This is a very difficult and general question, and many partial results are known. The paper of Brown, Erdős, and Sós [BES73] contains many of them, in particular the general lower bound, for all $k>r$ and $s>1$,\[\mathrm{ex}_r(n,\mathcal{F})\gg_{k,s} n^{\frac{rs-k}{s-1}}.\]A general conjecture of Brown, Erdős, and Sós is that, for all $r>t\geq 2$ and $s\geq 3$,\[\mathrm{ex}_t(n,\mathcal{F})=o(n^t)\]whenever $k\geq (r-t)s+t+1$. The case $t=2$ is the subject of [1178].

The case $s=r=3$ and $k=6$ is the subject of [716]. The case of $r=3$ and $k=s+2$ is the subject of [1076].

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This page was last edited 24 January 2026.