The optimal block-coloring conjecture for generalized Schur triples at parameter one-half

Let $n$ be a positive integer, and let $[n]=\{1,\ldots,n\}$. A generalized Schur triple is a triple of the form

$$\bigl(x,y,x+\lfloor y/2\rfloor\bigr).$$

A $2$-coloring of $[n]$ is written in block form as $R^sB^{t-s}R^{u-t}B^{n-u}$, where the exponents specify consecutive red and blue intervals. Let $M(n)$ denote the minimum number of monochromatic generalized Schur triples over all $2$-colorings of $[n]$.

The optimal block-coloring conjecture. For $n\geq12$,

$$M(n)=\left\lfloor\frac{n^2+5}{6}\right\rfloor,$$

and this minimum is attained by the coloring $R^sB^{t-s}R^{u-t}B^{n-u}$ with

$$s=\left\lfloor\frac{n+3}{6}\right\rfloor,\qquad t=\left\lfloor\frac{n+1}{2}\right\rfloor,\qquad u=\left\lfloor\frac{5n+3}{6}\right\rfloor.$$

The claim describes the proposed true minimum over all $2$-colorings, rather than only colorings with three blocks. The preceding restricted formula is not globally optimal for $0<a<1$, as explicit examples show; the conjecture gives the expected exact answer for $a=\frac12$ when $n\geq12$.