We study structural properties of graphs with bounded clique number and high minimum degree. In particular, we show that there exists a function L = L(r,ɛ) such that every Kr-free graph G on n vertices with minimum degree at least ((2r–5)/(2r–3)+ɛ)n is homomorphic to a Kr-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 Łuczak (2006) and, more recently, Goddard and Lyle (2011) deduced the general case from Łuczak’s result. Łuczak’s proof was based on an application of Szemerédi’s regularity lemma and, as a consequence, it only gave rise to a tower-type bound on L(3, ɛ). The proof presented here replaces the application of the regularity lemma by a probabilistic argument, which yields a bound for L(r, ɛ) that is doubly exponential in poly(ɛ).

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关于有界团数稠密图的结构

我们研究了有界团数和高最小度图的结构性质。特别是，我们证明存在一个函数大号=大号(r,ɛ) 使得每个ķr- 自由图G在n具有最小度数的顶点至少 ((2r–5)/(2r–3)+ε)n同态于 aķr- 最多有图大号顶点。众所周知，对于这个结果，所需的最小度数条件几乎是最好的。对于r= 3 该结果由 Łuczak (2006) 获得，最近，Goddard 和 Lyle (2011) 从 Łuczak 的结果中推断出一般情况。Łuczak 的证明是基于 Szemerédi 的正则引理的应用，因此，它只产生了大号(3,ε）。这里提出的证明用概率论证代替了正则引理的应用，这产生了大号(r,ε) 在 poly(ε）。