Variations on Ramsey numbers and minimum numbers of monochromatic triangles in line $2$-colorings of configurations

TL;DR

This paper investigates Ramsey numbers and monochromatic triangles in line 2-colorings of configurations, extending classical graph theory results to geometric configurations and analyzing their combinatorial properties.

Contribution

It introduces the concept of line 2-colorings of configurations, computes specific examples, and explores how configuration operations affect monochromatic triangle counts.

Findings

Computed minimal monochromatic triangles for symmetric configurations

Extended Ramsey theory to geometric configurations

Linked maximum intersecting lines to triangle minimization

Abstract

This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by defining line $2$-colorings of configurations of points and lines and considering the minimum number of non-disjoint monochromatic triangles. We compute specific examples for notable symmetric $v_{3}$ configurations before considering a general result regarding the addition or connected sum of configurations through incidence switches. The paper finishes by considering the maximal number of mutually intersecting lines and how this relates to the minimum number of triangles given a line $2$-coloring of a symmetric $v_{3}$ configuration.