Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. R\"{ o}dl, A. %Ruci\'{ n}ski, M. Simonovits, J. Skokan, The Ramsey number for hypergraph cycles I, J. Combin. Theory, Ser. A, 113 (2006), 67-83] proved that the 2-color Ramsey number of 3-uniform loose cycles on $2n$ vertices is asymptotically $\frac{ 5n}{ 2}$. Their proof is based on the method of 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 $n\geq m\geq 3$, $R(\mathcal{ P}^3_n,\mathcal{ P}^3_m)=R(\mathcal{ P}^3_n,\mathcal{ C}^3_m)=R(\mathcal{ C}^3_n,\mathcal{ C}^3_m)+1=2n+\lfloor\frac{ m+1}{ 2}\rfloor$ and for $n>m\geq3$, $R(\mathcal{ P}^3_m,\mathcal{ C}^3_n)=2n+\lfloor\frac{ m-1}{ 2}\rfloor$. These give a positive answer to a question of Gy\'{ a}rf\'{ a}s and Raeisi [The Ramsey number of loose triangles and quadrangles in hypergraphs, Electron. J. Combin. 19 (2012), #R30].
Categories: math.CO