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This research aims to determine exact Ramsey numbers for pairs of complete bipartite graphs.

August 12, 2026Open Access

Five Exact Ramsey Numbers for Pairs of Complete Bipartite Graphs

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Randomized analysis reveals five exact Ramsey numbers in bipartite graphs, indicating unresolved cases in combinatorial theory.

This research aims to determine exact Ramsey numbers for pairs of complete bipartite graphs.
Calculation of Ramsey numbers R(K_{2,17},K_{2,17}) to R(K_{2,47},K_{2,48}) using graph theory.
Application of existing theorems by Lortz and Mengersen and construction of symmetric matrices.
Inclusion of deterministic verification for five identified Ramsey numbers.
R(K_{2,17},K_{2,17})=66, R(K_{2,29},K_{2,30})=116, R(K_{2,39},K_{2,40})=156, R(K_{2,43},K_{2,44})=172, R(K_{2,47},K_{2,48})=188 were established with definitive values.
The research confirms that R(K_{2,n-1},K_{2,n})=4n-4 for every 3<=n<=58 as derived from established patterns.
The work highlights four asymmetric Ramsey cases as unresolved in the latest dynamic survey.

D'Ambrosio et al. (2026) studied this question.

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