Inverting the Turan problem with chromatic number

For a graph G and a family of graphs H, the Turan number ex(G, H) is defined to be the maximum number of edges among all H-free subgraphs of G. Inverting this problem, Briggs and Cox (2019) [5] studied the extremal function epsilon(H)(k) = sup{e(G) vertical bar ex(G, H) < k}, where e(G) is the size of G, and suggested to investigate the extremal function phi(H)(k) = sup{chi(G) vertical bar ex(G, H) < k}, where chi(G) denotes the chromatic number of G. Let K-n be a complete graph of order nand Ha given graph. In this paper, we establish a tight general upper bound for phi(H)(k) and conjecture phi(H)(k) = max{n vertical bar ex(K-n, H) < k} for H not equal 2K(2). We also confirm this conjecture for many instances of H. (C) 2021 Elsevier B.V. All rights reserved.