Almost Empty Monochromatic Quadrilaterals in Planar Point Sets


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For positive integers c, s ≥ 1, r ≥ 3, let Wr(c, s) be the least integer such that if a set of at least Wr(c, s) points in the plane, no three of which are collinear, is colored with c colors, then this set contains a monochromatic r-gon with at most s interior points. As is known, if r = 3, then Wr(c, s)=Wr(c, s). In this paper, we extend these results to the case r = 4. We prove that W4(2, 1) = 11, W4(3, 2) ≤ 120, and the least integer μ4(c) such that W4(c, μ4(c)) < ∞ is bounded by \(\left\lfloor {\frac{{c - 1}}{2}} \right\rfloor \cdot 2 \leqslant \mu 4\left( c \right) \leqslant 2c - 3\),where c ≥ 2.

作者简介

L. Liu

School of Mathematics

编辑信件的主要联系方式.
Email: liuling00ling@163.com
中国, Tianjin

Y. Zhang

School of Mathematics

Email: liuling00ling@163.com
中国, Tianjin

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