A simple proof of the Gan-Loh-Sudakov conjecture

Abstract

We give a new unified proof that any simple graph on $n$ vertices with maximum degree at most $\Delta$ has no more than $a\left(\genfrac{}{}{0ex}{}{\Delta +1}{t}\right)+\left(\genfrac{}{}{0ex}{}{b}{t}\right)$ cliques of size $t(t\ge 3)$, where $n=a(\Delta +1)+b(0\le b\le \Delta )$.

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• Many cliques in bounded-degree hypergraphs