A few remarks on Ramsey--Turán-type problems
For fixed graph \(H\) and positive integer function \(f(n)\), the Ramsey-Turán number \(\text{RT}(n, H,f(n))\) is the maximum number of edges in a graph \(G\) of order \(n\) without having a copy of \(H\) and without \(f(n)\) independent vertices. Some improved bounds for some Ramsey-Turán numbers are proved. For example, it is shown for \(r > 2\), \(\text{RT}(n, K_4, n^{1 - 1/r}) < n^{2-1/r(r+1)}\). Results such as ``if \(H\) is a graph whose vertices can be partitioned into two sets each inducing an acyclic graph, then \(\text{RT}(n, H, ne^{-\omega(n)\sqrt{\ln n}}) = o(n^2)\), where \(\omega(n) \rightarrow \infty\) arbitrarily slowly with \(n\)'' are proved and used to give bounds for specific graphs.
scientific article; zbMATH DE number 1970679
scientific article