Abstract
Let us defineG(n) to be the maximum numberm such that every graph onn vertices contains at leastm homogeneous (i.e. complete or independent) subgraphs. Our main result is exp (0.7214 log2 n) ≧G(n) ≧ exp (0.2275 log2 n), the main tool is a Ramsey—Turán type theorem.
We formulate a conjecture what supports Thomason’s conjecture\(\mathop {\lim }\limits_{k \to \infty } \) R(k, k)1/k = 2.
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