Ramsey numbers of 3-uniform loose paths and loose cycles

The 3-uniform loose cycle, denoted by C"n^3, is the hypergraph with vertices v"1,v"2,...,v"2"n and n edges v"1v"2v"3,v"3v"4v"5,...,v"2"n"-"1v"2"nv"1. Similarly, the 3-uniform loose pathP"n^3 is the hypergraph with vertices v"1,v"2,...,v"2"n"+"1 and n edges v"1v"2v"3,v"3v"4v"5,...,v"2"n"-"1v"2"nv"2"n"+"1. In 2006 Haxell et al. proved that the 2-color Ramsey number of 3-uniform loose cycles on 2n vertices is asymptotically 5n2. Their proof is based on the method of the Regularity Lemma. Here, without using this method, we generalize their result by determining the exact values of 2-color Ramsey numbers involving loose paths and cycles in 3-uniform hypergraphs. More precisely, we prove that for every n>=m>=3,R(P"n^3,P"m^3)=R(P"n^3,C"m^3)=R(C"n^3,C"m^3)[email protected][email protected]?, and for every n>m>=3, R(P"m^3,C"n^3)[email protected][email protected]?. This gives a positive answer to a recent question of Gyarfas and Raeisi.