Ramsey numbers of $4$-uniform loose cycles

Abstract: Gy\'arf\'as, S\'ark\"ozy and Szemer\'edi proved that the $2$-color Ramsey number $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ of a $k$-uniform loose cycle $\mathcal{C}^k_n$ is asymptotically $\frac{1}{2}(2k-1)n,$ generating the same result for $k=3$ due to Haxell et al. Concerning their results, it is conjectured that for every $n\geq m\geq 3$ and $k\geq 3,$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_m)=(k-1)n+\lfloor\frac{m-1}{2}\rfloor.$$ In $2014$, the case $k=3$ is proved by the authors. Recently, the authors showed that this conjecture is true for $n=m\geq 2$ and $k\geq 8$. In this paper, 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 $R(\mathcal{C}^4_n,\mathcal{C}^4_n)$ is between two values with difference one.