On Erdős's Conjecture on Multiplicities of Complete Subgraphs Lower Upper Bound for Cliques of Size 6

, and let ct = limn→∞ ct(n). An old conjecture of Erdös, related to Ramsey’s theorem, states that ct = 2 1−(t 2 ). It was shown false by Thomason for all t ≥ 4 ([3],[4]). Franek and Rödl ([1]) presented a simpler counterexample to the conjecture for t = 4 derived from a simple Cayley graph of order 2 obtained by a computer search giving essentially the same upper bound for c4 as Thomason’s. In this note we show that the same graph gives rise to two sequences of graphs, one a counteraxmple for t = 5 and the other for t = 6 improving the original Thomason’s c5 < 0.906·2 −9 to c5 ≤ 0.885834·2 −9 (though Jagger, Thomason, and Šťov́ıček [2] obtained a better c5 ≤ 0.8801·2 ), and Thomason’s