On the structure of dense graphs with fixed clique number

We study structural properties of graphs with fixed clique number and high minimum degree. In particular, we show that there exists a function $L=L(r,\varepsilon)$, such that every $K_r$-free graph $G$ on $n$ vertices with minimum degree at least $(\frac{2r-5}{2r-3}+\varepsilon)n$ is homomorphic to a $K_r$-free graph on at most $L$ vertices. It is known that the required minimum degree condition is approximately best possible for this result. For $r=3$ this result was obtained by \L uczak [On the structure of triangle-free graphs of large minimum degree, Combinatorica 26 (2006), no. 4, 489-493] and, more recently, Goddard and Lyle [Dense graphs with small clique number, J. Graph Theory 66 (2011), no. 4, 319-331] deduced the general case from \L uczak's result. \L uczak's proof was based on an application of Szemer\'edi's regularity lemma and, as a consequence, it only gave rise to a tower-type bound on $L(3,\varepsilon)$. The proof presented here replaces the application of the regularity lemma by a probabilistic argument, which yields a bound for $L(r,\varepsilon)$ that is doubly exponential in poly($\varepsilon$).