Off-diagonal hypergraph Ramsey numbers

The Ramsey number rk(s, n) is the minimum N such that every red-blue coloring of the k-subsets of {1,..., N} contains a red set of size s or a blue set of size n, where a set is red (blue) if all of its k-subsets are red (blue). A k-uniform tight path of size s, denoted by Ps, is a set of s vertices v1 < · · · < vs in Z, and all s−k+1 edges of the form {vj, vj+1,..., vj+k−1}. Let rk(Ps, n) be the minimum N such that every red-blue coloring of the k-subsets of {1,..., N} results in a red Ps or a blue set of size n. The problem of estimating both rk(s, n) and rk(Ps, n) for k = 2 goes back to the seminal work of Erdős and Szekeres from 1935, while the case k ≥ 3 was first investigated by Erdős and Rado in 1952. In this paper, we deduce a quantitative relationship between multicolor variants of rk(Ps, n) and rk(n, n). This yields several consequences including the following: • We determine the correct tower growth rate for both rk(s, n) and rk(Ps, n) for s ≥ k + 3. The question of determining the tower growth rate of rk(s, n) for all s ≥ k + 1 was posed by Erdős and Hajnal in 1972. • We show that determining the tower growth rate of rk(Pk+1, n) is equivalent to determining the tower growth rate of rk(n, n), which is a notorious conjecture of Erdős, Hajnal and Rado from 1965 that remains open. Some related off-diagonal hypergraph Ramsey problems are also explored.