Exact Lower Bounds for Monochromatic Schur Triples and Generalizations

We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots ,n\}$, subject to a coloring with two different colors. Previously, only asymptotic formulas for such bounds were known, and only for $a\in \mathbb{N}$. Using symbolic computation techniques, these results are extended here to arbitrary $a\in \mathbb{R}$. Furthermore, we give exact formulas for the minimum number of monochromatic Schur triples for $a=1,2,3,4$, and briefly discuss the case $0<a<1$.