2024/11/23 by Suk, Andrew, Zeng, Ji
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.15649
The monotone path Pn+2 is an ordered 3-uniform hypergraph whose vertex set has size n+2 and edge set consists of all consecutive triples. In this note, we consider the collection Jn of ordered 3-uniform hypergraphs named monotone paths with n jumps, and we prove the following relation r(3;n) ≤ R(Pn+2,Jn) ≤ 4n ⋅ r(3;n), where r(3;n) is the multicolor Ramsey number for triangles and R(Pn+2,Jn) is the hypergraph Ramsey number for Pn+2 versus any member of Jn. In particular, whether r(3;n) is exponential, which is a very old problem of Erdős, is equivalent to whether R(Pn+2,Jn) is exponential.