Bipartite Ramsey numbers involving stars, stripes and trees

Michalis Christou (King's College London)
Costas Iliopoulos (King's College London)
Mirka Miller (University of Newcastle)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 1, No 2 (2013): Electronic Journal of Graph Theory and Applications

The Ramsey number R(m, n) is the smallest integer p such that any blue-red colouring of the edges of the complete graph Kp forces the appearance of a blue Km or a red Kn. Bipartite Ramsey problems deal with the same questions but the graph explored is the complete bipartite graph instead of the complete graph. We consider special cases of the bipartite Ramsey problem. More specifically we investigate the appearance of simpler monochromatic graphs such as stripes, stars and trees under a 2-colouring of the edges of a bipartite graph. We give the Ramsey numbers Rb(mP2, nP2), Rb(Tm, Tn), Rb(Sm, nP2), Rb(Tm, nP2) and Rb(Sm, Tn).