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- W2952185123 abstract "For positive integers $c, s geq 1$, let $M_3(c, s)$ be the least integer such that any set of at least $M_3(c, s)$ points in the plane, no three on a line and colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. The case $s=0$, which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that $M_3(1, 0)=3$, $M_3(2, 0)=9$ and $M_3(c, 0)=infty$, for $cgeq 3$. In this paper we extend these results when $c geq 2$ and $s geq 1$. We prove that the least integer $lambda_3(c)$ such that $M_3(c, lambda_3(c))< infty$ satisfies: $$leftlfloorfrac{c-1}{2}rightrfloor leqlambda_3(c)leq c-2,$$ where $c geq 2$. Moreover, the exact values of $M_3(c, s)$ are determined for small values of $c$ and $s$. We also conjecture that $lambda_3(4)=1$, and verify it for sufficiently large Horton sets." @default.
- W2952185123 created "2019-06-27" @default.
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- W2952185123 date "2014-10-02" @default.
- W2952185123 modified "2023-09-27" @default.
- W2952185123 title "Almost Empty Monochromatic Triangles in Planar Point Sets" @default.
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