Maximum chromatic number of the complement graph of graphs with fixed \(h(G)\)

In [1], Erdös raised the following question:

Problem [1]

For each integer \( t \geq 6\), what is the minimum value \( c\) such that any graph without isolated vertices having \( h(G) \leq t\) satisfies \( \chi(\bar{G}) \leq n^c\)?

1 P. Erdös, On the covering of the vertices of a graph by cliques, J. Math. Res. Exposition 2 (1982) no. 1, 9396.