 
Summary: Asymptotically tight bounds for some multicolored Ramsey
numbers
Noga Alon
Vojtech R¨odl
Abstract
Let H1, H2, . . . , Hk+1 be a sequence of k + 1 finite, undirected, simple graphs. The (mul
ticolored) Ramsey number r(H1, H2, . . . , Hk+1) is the minimum integer r such that in every
edgecoloring of the complete graph on r vertices by k + 1 colors, there is a monochromatic
copy of Hi in color i for some 1 i k + 1. We describe a general technique that supplies
tight lower bounds for several numbers r(H1, H2, . . . , Hk+1) when k 2, and the last graph
Hk+1 is the complete graph Km on m vertices. This technique enables us to determine the
asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In
particular we show that r(K3, K3, Km) = (m3
poly log m), thus solving (in a strong form) a
conjecture of Erdos and S´os raised in 1979. Another special case of our result implies that
r(C4, C4, Km) = (m2
poly log m) and that r(C4, C4, C4, Km) = (m2
/ log2
m). The proofs
combine combinatorial and probabilistic arguments with spectral techniques and certain esti
