On the Multicolor Ramsey Number for 3-Paths of Length Three

We show that if we color the hyperedges of the complete 3-uniform hypergraph on 2n + √ 18n + 1 + 2 vertices with n colors, then one of the color classes contains a loose path of length three. Let P denote the 3-uniform path of length three by which we mean the only connected 3-uniform hypergraph on seven vertices with the degree sequence (2, 2, 1, 1, 1, 1, 1). By R(P ;n) we denote the multicolored Ramsey number for P defined as the smallest number N such that each coloring of the hyperedges of the complete 3-uniform hypergraph K (3) N with n colors leads to a monochromatic copy of P . It is easy to check that R(P ;n) > n+6 (see [2, 5]), and it is believed that in fact equality holds, i.e.