Abstract

Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed to find asymptotic results for very large graphs, so it seems that the method is not suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove the exact values $R(K_4^-,K_4^-,K_4^-)=28$, $R(K_8,C_5)= 29$, $R(K_9,C_6)= 41$, $R(Q_3,Q_3)=13$, $R(K_{3,5},K_{1,6})=17$, $R(C_3, C_5, C_5)= 17$, and $R(K_4^-,K_5^-;3)= 12$. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.

Keywords

  1. flag algebra
  2. Ramsey numbers
  3. semidefinite programming
  4. small graphs

MSC codes

  1. 05C35
  2. 05C55
  3. 05D10

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 2328 - 2344
ISSN (online): 1095-7146

History

Submitted: 7 February 2018
Accepted: 13 June 2021
Published online: 6 October 2021

Keywords

  1. flag algebra
  2. Ramsey numbers
  3. semidefinite programming
  4. small graphs

MSC codes

  1. 05C35
  2. 05C55
  3. 05D10

Authors

Affiliations

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-1600390, DMS-1855653
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1600483, DMS-1855622
National Science Foundation https://doi.org/10.13039/100000001 : CNS 1229081, 1205413, AGS 0835579, CNS 0958354, CNS-0821794

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