A Ramsey-type Theorem on the Max-Cut Value of d-Regular Graphs.

In this paper we study the problem of finding large cuts in $K_r$-free graphs with max degree $d$. We show that such graphs always have cuts which cut a $1/2+Omega(1/d^{1-1/d^{r-2}})$ fraction of edges. This generalizes known results for $K_3$-free graphs. A key component of this result is showing that graphs with few triangles also have non-trivially good cuts.