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An exact value for the Ramsey number $R(K_5, K_{5-e})$

Vigleik Angeltveit

TL;DR

The paper resolves the exact Ramsey number $R(5,4.5)$, proving it equals $30$ by combining a census of smaller Ramsey graphs, linear programming constraints via the edge equation, a gluing framework to assemble potential $\mathcal{R}(5,4.5)$ graphs from $\mathcal{R}(5,3.5)$ and $\mathcal{R}(4,4.5)$, and SAT-based verifications. It develops and implements two complementary computational pipelines: a gluing-based search with feasible cones and collapsing rules, and a SAT-encoded enumeration of gluing configurations; both are reinforced by a detailed census and extensive graph counting across degree splits. The results exhaustively show the nonexistence of $\mathcal{R}(5,4.5,30)$ graphs, thereby establishing $R(5,4.5)=30$ and refining the computational frontier for small Ramsey numbers. The work demonstrates a robust methodology for tackling exact Ramsey numbers via hybrid combinatorial and SAT/constraint-solving techniques, with implications for future precision results such as $R(5,4.5)$-level cases and related half-integer Ramsey numbers.

Abstract

We compute the exact value of the Ramsey number $R(K_5, K_{5-e})$. It is equal to 30.

An exact value for the Ramsey number $R(K_5, K_{5-e})$

TL;DR

The paper resolves the exact Ramsey number , proving it equals by combining a census of smaller Ramsey graphs, linear programming constraints via the edge equation, a gluing framework to assemble potential graphs from and , and SAT-based verifications. It develops and implements two complementary computational pipelines: a gluing-based search with feasible cones and collapsing rules, and a SAT-encoded enumeration of gluing configurations; both are reinforced by a detailed census and extensive graph counting across degree splits. The results exhaustively show the nonexistence of graphs, thereby establishing and refining the computational frontier for small Ramsey numbers. The work demonstrates a robust methodology for tackling exact Ramsey numbers via hybrid combinatorial and SAT/constraint-solving techniques, with implications for future precision results such as -level cases and related half-integer Ramsey numbers.

Abstract

We compute the exact value of the Ramsey number . It is equal to 30.
Paper Structure (18 sections, 1 theorem, 6 equations, 5 tables)

This paper contains 18 sections, 1 theorem, 6 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. A census of $\mathcal{R}(4,4.5)$ and $\mathcal{R}(5,3.5)$
  4. Some linear programming
  5. Gluing
  6. Encoding a gluing problem as a SAT problem
  7. Detailed graph counts
  8. $(d_1, d_2) = (14, 15)$
  9. $(d_1, d_2) = (15, 14)$
  10. $(d_1, d_2) = (16, 13)$
  11. $(d_1, d_2) = (17, 12)$
  12. $(d_1, d_2) = (18, 12)$
  13. The first method
  14. $(d_1, d_2) = (14, 15)$
  15. $(d_1, d_2) = (15, 14)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

The Ramsey number $R(5,4.5)$ is equal to $30$.

Theorems & Definitions (1)

  • Theorem 1.1