2014/10/02 by Deepan Basu, Basu, Deepan, Kinjal Basu +5
Computer Science · Mathematics · Social Sciences · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Point processes and geometric inequalities #Urbanization and City Planning
paper · pdf · doi:10.48550/arxiv.1410.0424
openalex publication_date 2014/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For positive integers c, s ≥ 1, let M3(c, s) be the least integer such that any set of at least M3(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 M3(1, 0)=3, M3(2, 0)=9 and M3(c, 0)=∞, for c≥ 3. In this paper we extend these results when c ≥ 2 and s ≥ 1. We prove that the least integer λ3(c) such that M3(c, λ3(c))< ∞ satisfies: \lfloor(c-1)/(2)\rfloor ≤λ3(c)≤ c-2, where c ≥ 2. Moreover, the exact values of M3(c, s) are determined for small values of c and s. We also conjecture that λ3(4)=1, and verify it for sufficiently large Horton sets.