Edge Distribution of Graphs with Few Copies of a Given Graph

Abstract

We show that if a graph contains few copies of a given graph, then its edges are distributed rather unevenly.

In particular, for all $\varepsilon > 0$ and $r\geq2$, there exist $\xi =\xi (\varepsilon,r) > 0$ and $k=k (\varepsilon,r)$ such that, if $n$ is sufficiently large and $G=G(n)$ is a graph with fewer than $\xi n^{r}$$r$-cliques, then there exists a partition $V(G) =\cup_{i=0}^{k}V_{i}$ such that \[ \vert V_{i}\vert =\lfloor n/k\rfloor \quad \text{and} \quad e(W_{i}) <\varepsilon\vert V_{i}\vert ^{2}\] for every $i\in [k]$.

We deduce the following slightly stronger form of a conjecture of Erdős.

For all $c>0$ and $r\geq3$, there exist $\xi=\xi (c,r) >0$ and $\beta=\beta(c,r)>0$ such that, if $n$ is sufficiently large and $G=G(n,\lceil cn^{2} \rceil)$ is a graph with fewer than $\xi n^{r}$$r$-cliques, then there exists a partition $V(G) =V_{1}\cup V_{2}$ with $ \vert V_{1} \vert = \lfloor n/2 \rfloor $ and $\vert V_{2} \vert = \lceil n/2 \rceil $ such that \[ e(V_{1},V_{2}) > (1/2+\beta) e (G).\]