On the chromatic number of subgraphs of a given graph

Abstract

It is shown that for arbitrary positive integers m,n there exists a ϕ ( m , n ) \phi (m,n) such that if χ ( G ) ⩾ ϕ ( m , n ) \chi (G) \geqslant \phi (m,n) , then G contains either a complete subgraph of size m or a subgraph H with χ ( H ) = n \chi (H) = n containing no C 3 {C_3} . This gives an answer to a problem of Erdös and Hajnal.