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 1 v 2 v 3 , v 3 v 4 v 5 , … , v 2 n − 1 v 2 n v 1 . Similarly, the 3-uniform loose path P n 3 is the hypergraph with vertices v 1 , v 2 , … , v 2 n + 1 and n edges v 1 v 2 v 3 , v 3 v 4 v 5 , … , v 2 n − 1 v 2 n v 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 5 n 2 . 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 ) + 1 = 2 n + ⌊ m + 1 2 ⌋ , and for every n > m ⩾ 3 , R ( P m 3 , C n 3 ) = 2 n + ⌊ m − 1 2 ⌋ . This gives a positive answer to a recent question of Gyárfás and Raeisi.