Steinberg, Richard; Tovey, Craig A. Planar Ramsey numbers. (English) Zbl 0794.05091 J. Comb. Theory, Ser. B 59, No. 2, 288-296 (1993). The planar Ramsey number \(\text{PR}(k,\ell)\) \((k,\ell\geq 2)\) is the smallest integer \(n\) such that any planar graph on \(n\) vertices contains either a complete graph on \(k\) vertices or an independent set of size \(\ell\). We find exact values of \(\text{PR}(k,\ell)\) for all \(k\) and \(\ell\). Included is a proof of a 1976 conjecture due to Albertson, Bollobás, and Tucker that every triangle-free planar graph on \(n\) vertices contains an independent set of size \(\bigl\lfloor{n\over 3}\bigr\rfloor+1\). Reviewer: R.Steinberg (Murray Hill) Cited in 3 ReviewsCited in 27 Documents MSC: 05C55 Generalized Ramsey theory Keywords:planar Ramsey number; planar graph; complete graph; independent set PDFBibTeX XMLCite \textit{R. Steinberg} and \textit{C. A. Tovey}, J. Comb. Theory, Ser. B 59, No. 2, 288--296 (1993; Zbl 0794.05091) Full Text: DOI