Ramsey numbers of 3-uniform loose paths and loose cycles

Original · EN

Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. Rödl, A. %Ruciński, 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 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≥ m≥ 3, R(P³ₙ,P³ₘ)=R(P³ₙ,C³ₘ)=R(C³ₙ,C³ₘ)+1=2n++1/2 and for n>m≥3, R(P³ₘ,C³ₙ)=2n+-1/2. These give a positive answer to a question of Gyárfás and Raeisi [The Ramsey number of loose triangles and quadrangles in hypergraphs, Electron. J. Combin. 19 (2012), #R30].