Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold
David Conlon, Jacob Fox, Huy Tuan Pham, Yufei Zhao

TL;DR
This paper investigates the growth of set-coloring Ramsey numbers near a critical ratio of colors, establishing bounds by linking them to error-correcting codes close to the zero-rate threshold.
Contribution
It provides new bounds for set-coloring Ramsey numbers in the intermediate regime by connecting them to error-correcting codes near the zero-rate threshold.
Findings
Bounds for $R(n;r,s)$ near the critical ratio are established.
Connection between Ramsey numbers and error-correcting codes is demonstrated.
Results clarify the growth behavior of these numbers in the intermediate range.
Abstract
For positive integers with , the set-coloring Ramsey number is the minimum such that if every edge of the complete graph receives a set of colors from a palette of colors, then there is a subset of vertices where all of the edges between them receive a common color. If is fixed and is less than and bounded away from , then is known to grow exponentially in , while if is greater than and bounded away from , then is bounded. Here we prove bounds for in the intermediate range where is close to by establishing a connection to the maximum size of error-correcting codes near the zero-rate threshold.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Complexity and Algorithms in Graphs · Computability, Logic, AI Algorithms
