Ramsey Numbers of Connected Clique Matchings
Barnaby Roberts

TL;DR
This paper establishes the exact Ramsey number for connected clique matchings in 2-edge-coloured complete graphs, showing a precise threshold for the existence of a monochromatic connected subgraph with multiple disjoint cliques.
Contribution
The authors determine the exact Ramsey number for connected clique matchings, providing a tight bound that was previously unknown.
Findings
The Ramsey number for connected clique matchings is exactly (r^2 - r - 1)n - r + 1.
Below this number, such a monochromatic connected subgraph always exists.
The bound is tight and cannot be improved.
Abstract
We determine the Ramsey number of a connected clique matching. That is, we show that if is a -edge-coloured complete graph on vertices, then there is a monochromatic connected subgraph containing disjoint copies of , and that this number of vertices cannot be reduced.
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