Ramsey numbers of 4-uniform loose cycles

Gyárfás, Sárközy and Szemerédi proved that the 2-color Ramsey number R(Cnk,Cnk) of a k-uniform loose cycle Cnk is asymptotically 12(2k−1)n, generating the same result for k=3 due to Haxell et al. Concerning their results, it is conjectured that for every n≥m≥3 and k≥3, R(Cnk,Cmk)=(k−1)n+⌊m−12⌋.In 2014, the case k=3 is proved by the authors. Recently, the authors showed that this conjecture is true for n=m≥2 and k≥8. Their method can be used for case n=m≥2 and k=7, but more details are required. The only open cases for the above conjecture when n=m are k=4,5,6. Here, we investigate the case k=4, and we show that the conjecture holds for k=4 when n>m or n=m is odd. When n=m is even, we show that R(Cn4,Cn4) is between two values with difference one.