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

Jamie Bishop (St. Martin's University, Lacey WA, USA)
Rebekah Kuss (Oregon State University$^{1}$, Corvallis OR, USA)
Benjamin Peet (St. Martin's University, Lacey WA, USA)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 11, No 2 (2023): Electronic Journal of Graph Theory and Applications

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 v3 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 v3 configuration.