Exact lower bounds for monochromatic Schur triples and generalizations

Christoph Koutschan and Elaine Wong

Graph of the area function at local minima


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 $0< a< 1$.


The material on this webpage accompanies the article Exact lower bounds for monochromatic Schur triples and generalizations by Christoph Koutschan and Elaine Wong.

A triple $(x,y,z)\in\mathbb{N}^3$ is called a Schur triple if its entries satisfy the equation $x+y=z$. A coloring of the set $\{1,\dots,n\}$ is a map that assigns to each element a color (red or blue). We say that a Schur triple is monochromatic (with respect to a given coloring) if all of its entries have the same color. We are interested in the coloring that produces the least number of monochromatic Schur triples. For this purpose, we study colorings of the form $R^sB^{t-s}R^{n-t}$, i.e., the first $s$ numbers are red, the numbers $\{s+1,\dots,t\}$ are blue, and the remaining numbers are again red. Below the situation is depicted for $n=44$, and for the optimal choice $s=16$ and $t=40$. Each monochromatic triple $(x,y,x+y)$ is represented by a dot at position $(x,y)$. The shaded regions indicate the location of non-monochromatic triples.

Monochromatic Schur triples

The following animation shows the case of generalized Schur triples $(x,y,x+\lfloor ay\rfloor)$ where $a$ is a positive real number. Here the optimal choice for $s$ and $t$ also depends on the value of $a$. The colored regions show the location of monochromatic triples, so that their area is related to their total number. For each $a$, the values for $s,t$ have been chosen as to minimize this area. They are visible as horizontal and vertical lines in the picture.

Optimal choice for s and t

Supplementary electronic material

We provide a Mathematica notebook containing all computations that constitute the proofs in our article, and the formulas of all relevant quantities in Mathematica format. Detailed explanations in the notebook are given in order to help the reader understand the technical details of the computations.