Induced cycles and chromatic number (Q1306304)

The author proves that for any pair of positive integers \(r\) and \(s\), there exists an integer \(N(r,s)\) such that every graph with chromatic number at least \(N(r,s)\) contains either the complete graph \(K_r\) or an induced odd cycle of length at least 5 or an induced cycle of length at least \(s\). This result is a weakened version of two conjectures made by A. Gyárfás (1985) which are related to the well-known strong perfect graph conjecture as well as the Gyárfás-Sumner conjecture (1973 and 1981). Several similar/related problems had been studied before by A. Gyárfás, H. A. Kierstead, S. G. Penrice, D. P. Sumner, E. Szemeredi, W. T. Trotter and Zs. Tuza.