On the Minimum Number of Monochromatic Generalized Schur Triples

Abstract

The solution to the problem of finding the minimum number of monochromatic triples (x, y, x + ay) with a 2 being a fixed positive integer over any 2-coloring of [1, n] was conjectured by Butler, Costello, and Graham (2010) and Thanathipanonda (2009). We solve this problem using a method based on Datskovsky's proof (2003) on the minimum number of monochromatic Schur triples (x, y, x + y). We do this by exploiting the combinatorial nature of the original proof and adapting it to the general problem.

Exact Lower Bounds for Monochromatic Schur Triples and Generalizations

“…without the requirement of x ≤ y. This was conjectured by Thanatipanonda [13] and Butler, Costello, and Graham [1], and subsequently proven in 2017 by Thanatipanonda and Wong [14].…”

“…First, we would like to note that Lemma 4 explains why the asymptotic formula for MGSTs for integral a ≥ 2 given in [1,13,14] does not specialize to the previously known case a = 1: this phenomenon is due to the piecewise definition of m(a), with a transition at 1 < α 8 < 2. Geometrically speaking, α 8 marks the point where the polygon 133 (see Figure 3) disappears, when a increases from 1 to 2, and s = s 0 (a) and t = t 0 (a) are updated constantly.…”

“…We now turn to generalized Schur triples, i.e., triples (x, y, z) subject to z = x + ay for some parameter a ∈ N, as studied by Thanatipanonda and Wong [14]. Here, we allow a to be even more general, i.e., a ∈ R + .…”

Avoiding Monochromatic Solutions to 3-term Equations

“…By symmetry, equations with a > b, c and b = c are taken care of as well. To summarize, all equations satisfying one of the following are uncommon: • a = c = 1 (b = 1: by [10], b > 1: by [13]).…”

“…. } [10,13] and x + y = z + w, where it turns out the 1 8 fraction of monochromatic solutions from a random coloring is asymptotically optimal (see Appendix A for proof). Asymptotic minima are studied most often, but there are also some results on exact minima [8].…”