On some problems of Euclidean Ramsey theory
Ilya D. Shkredov
TL;DR
This paper investigates Euclidean Ramsey theory, proving that any measurable two-coloring of the plane contains a monochromatic triangle with specific side restrictions, and explores analogous problems in finite fields.
Contribution
It establishes new results on monochromatic triangles in Euclidean plane colorings and extends the analysis to finite field settings.
Findings
Existence of monochromatic triangles with certain side restrictions in measurable plane colorings
Extension of Euclidean Ramsey problems to finite fields
New theoretical insights into geometric coloring problems
Abstract
In the paper we prove, in particular, that for any measurable coloring of the euclidian plane into two colours there is a monochromatic triangle with some restrictions on the sides. Also we consider similar problems in finite fields settings.
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