Local and global clique numbers (Q1328379)

A graph \(G\) is said to have the \((p,q)\)-property for some integers \(p\geq q\geq 2\) if for every \(p\)-set of its vertices the induced subgraph contains a \(q\)-clique. The aim of the paper is to investigate relations of the type \((p,q)\to (n,s)\), meaning that each graph having the \((p,q)\)- property also has the \((n,s)\)-property. Using matching methods it is proved that \((p,q)\to (n,n- p+ q)\) for \(p\leq 2q- 2\) and \(n\geq p- q+2\). The paper presents further results of that type using Ramsey and Turán numbers and also asymptotic results.