Ramsey numbers for tournaments

AbstractThe Ramsey number r(D1,…,Dk) of acyclic directed graphs D1,…,Dk is defined as the largest integer r for which there exists a tournament T=(V,A) on r vertices with a k-coloring ϕ:A→{1,…,k} of the arc set A such that no Di occurs in color i for any i∈{1,…,k}. We discuss recursive techniques to compute r(D1,…,Dk) in the case where there are paths and/or stars among the Di. In particular, solving a problem of Bialostocki and Dierker [Congr. Numer. 47 (1985) 119–123], we prove that r(D1,D2)=r(D1)·r(D2) holds if D1 is transitive and D2=Sn is an out-going star on n vertices. Our main result is an asymptotic formula for r(D1,…,Dk,Sn) where the digraphs D1,…,Dk are fixed arbitrarily and n→∞