The Ramsey number r ( H ; l ) is the minimum such that every -coloring of the -tuples of an -element set contains a monochromatic copy of H . For , and also in the case , we use . One of the initial achived bounds for hypergraph Ramsey numbers is a theorem due to Erd?s and Rado. They showed that where . In , Sudakov et al. presented two ways to improve an approuch due to Erd?s and Rado to give some upper bounds for the Ramsey number of complete 3-uniform hypergraphs. The first way was based on “builder and painter” game. In 2011, Sudakov et al. proposed an upper bound on -color Ramsey number of complete -partite 3-uniform hypergraph and consequently they answered to two questions of Erd?s an Hajnal.
