Theoretical Computer Science ( IF 0.747 ) Pub Date : 2020-08-26 , DOI: 10.1016/j.tcs.2020.08.019
Yan Li; Yusheng Li; Ye Wang

For graphs F, G and H, let $F\to \left(G,H\right)$ signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number $R\left(G,H\right)$ is defined as $\mathrm{min}\left\{r\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}{K}_{r}\to \left(G,H\right)\right\}$. In this note, we consider an optimization problem to bound the complete bipartite-critical Ramsey number ${R}_{\mathrm{\Lambda }}\left(G,H\right)$ defined as $\mathrm{max}\left\{t\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}{K}_{r}\setminus {K}_{t,t}\to \left(G,H\right)\right\}$ where $r=R\left(G,H\right)$ and Λ is a set of ${K}_{t,t}$, and determine ${R}_{\mathrm{\Lambda }}\left(G,H\right)$ for some pairs $\left(G,H\right)$.

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