Maximum set of edges no two covered by a clique (Q1084407)

Let h(G) be the largest number of edges of the graph G, no two of which are contained in the same clique. For G without isolated vertices it is proved that if \(h(G)\leq 5\), then \(\chi(\bar G)\leq h(G)\), but if \(h(G)=6\) then \(\chi(G)\) can be arbitrarily large, where \(\bar G\) is the complement of the graph G and \(\chi(\bar G)\) is its chromatic number.