Off-diagonal hypergraph Ramsey numbers
Dhruv Mubayi, Andrew Suk

TL;DR
This paper establishes the growth rates of certain hypergraph Ramsey numbers involving paths and off-diagonal cases, connecting these to longstanding open problems and conjectures in combinatorics.
Contribution
It determines the tower growth rates for off-diagonal hypergraph Ramsey numbers involving paths and links these to open conjectures, advancing understanding in hypergraph Ramsey theory.
Findings
Determined the tower growth rate for r_k(s,n) and r_k(P_s, n) for s ≥ k+3.
Showed equivalence between growth rates of r_k(P_{k+1}, n) and r_k(n,n).
Connected off-diagonal hypergraph Ramsey problems to longstanding conjectures.
Abstract
The Ramsey number is the minimum such that every red-blue coloring of the -subsets of contains a red set of size or a blue set of size , where a set is red (blue) if all of its -subsets are red (blue). A -uniform \emph{tight path} of size , denoted by , is a set of vertices in , and all edges of the form . Let be the minimum such that every red-blue coloring of the -subsets of results in a red or a blue set of size . The problem of estimating both and for goes back to the seminal work of Erdos and Szekeres from 1935, while the case was first investigated by Erdos and Rado in 1952. In this paper, we deduce a quantitative relationship between multicolor…
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Taxonomy
TopicsAdvanced Topology and Set Theory
