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The pigeonhole principle and multicolor Ramsey numbers

Vishal Balaji, Powers Lamb, Andrew Lott, Dhruv Patel, Alex Rice, Sakshi Singh and Christine Rose Ward

Vol. 15 (2022), No. 5, 857–884
Abstract

For integers k,r 2, the diagonal Ramsey number Rr(k) is the minimum N such that every r-coloring of the edges of a complete graph on N vertices yields a monochromatic subgraph on k vertices. Here we make a careful effort of extracting explicit upper bounds for Rr(k) from the pigeonhole principle alone. Our main term improves on previously documented explicit bounds for r 3, and we also consider an often-ignored secondary term, which allows us to subtract a positive proportion of the main term that is uniformly bounded below. Asymptotically, we give a self-contained proof that

Rr(k) (3 + e 2 ) (r(k 2))! ((k 2)!)r(1 + or(1)),

and we conclude by noting that our methods combine with previous estimates on Rr(3) to improve the constant 1 2(3 + e) to 1 2(3 + e) 1 48d, where d = 66 R4(3) 4. We also compare our formulas, and previously documented formulas, to some collected numerical data.

Keywords
Ramsey theory, Ramsey number, party problem, pigeonhole principle
Mathematical Subject Classification
Primary: 05C15
Milestones
Received: 7 December 2021
Revised: 17 February 2022
Accepted: 22 February 2022
Published: 3 March 2023

Communicated by Anant Godbole
Authors
Vishal Balaji
Department of Mathematics
Millsaps College
Jackson, MS
United States
Powers Lamb
Department of Mathematics
Millsaps College
Jackson, MS
United States
Andrew Lott
Department of Mathematics
Millsaps College
Jackson, MS
United States
Dhruv Patel
Department of Mathematics
Millsaps College
Jackson, MS
United States
Alex Rice
Department of Mathematics
Millsaps College
Jackson, MS
United States
Sakshi Singh
Department of Mathematics
Millsaps College
Jackson, MS
United States
Christine Rose Ward
Department of Mathematics
Millsaps College
Jackson, MS
United States