On Turan's theorem for sparse graphs
Let \(\alpha\) be the (vertex) independence number and let \(\log x=\max\{1,\ell nx\}\). Denote by \(f(n,t,p)\) the largest integer such that every graph of order \(n\) and average degree \(t\geq 1\) that contains no \(K_p\) satisfies \(\alpha\geq f(n,t,p)\). Theorem: There exists an absolute constant \(c_1\) such that \(f(n,t,p)>c_1\cdot(n/t)\cdot\log(\log t)/p\). This improves on the known bounds \(\alpha\geq n/(t+1)\) and \(\alpha>0.01(n/t)\log t\). The lasr inequality may be rewritten as \(f(n,t,3)>c\cdot(n/t)\log t\), and suggests the study of the question \(f(n,t,p)=c_p(n/t)\log t\).
scientific article; zbMATH DE number 3771639