We provide new values of the bipartite Ramsey number $R_B(C_4,K_{1,n})$ using induced subgraphs of the incidence graph of a projective plane. The approach, based on deleting subplanes of projective planes, has been used in related extremal problems and allows us to unify previous results and extend them. More importantly, using deep stability results on $2\mathrm{mod}p$ sets and double blocking sets, we can show some of the limits of this technique when the projective plane is Desarguesian of large enough square order. Finally, we also disprove two conjectures about $R_B(C_4,K_{1,n})$.

中文翻译：

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四周期相对于恒星的二分Ramsey数的新值

我们提供了二分Ramsey数的新值 ${\mathrm{[R}}_{乙}（C_4，{ķ}_{\mathrm{1个}，ñ}）$使用投影平面的入射图的归纳子图。该方法基于删除投影平面的子平面，已用于相关的极值问题，并使我们能够统一先前的结果并进行扩展。更重要的是，在$2\mathrm{模}p$集和双阻塞集，当投影平面是足够大的正方形阶的Desarguesian时，我们可以证明该技术的一些局限性。最后，我们还反驳了关于${\mathrm{[R}}_{乙}（C_4，{ķ}_{\mathrm{1个}，ñ}）$。