Paired endpoint neighbourhoods: $P_i$ has $u$ points, is closed under $x \mapsto 2m - x$ without fixed points, and its differences avoid two colours

This describes the endpoint neighbourhoods at the frontier with $k + 2$ colours: each one is a symmetric set of $u$ points whose differences avoid two colours.

Let $k, u, t, m \in \mathbb{N}$ with $2t = (k+1)u$ and $m = (k+2)t$, and suppose that $\mathrm{TR}(k, u + 1)$ holds, that is, every colouring with at most $k$ colours of the pairs of a finite set of at least $u + 1$ naturals has a monochromatic triangle ($R_k(3) \le u + 1$). Let $c : \mathbb{N} \to \mathrm{Fin}(k+2)$ be a Schur colouring of $[1, 2m + 1]$ with $k + 2$ colours, write $q = c(m+1)$, let $V_m = \{\, x \in [0, 2m+1] : x \ne m,\ c(|m - x|) = q \,\}$ be the central neighbourhood, and for a colour $i$ let

be the endpoint neighbourhood of colour $i$. If $i \ne q$, then:

1. $|P_i| = u$;
2. for every $x \in P_i$: $2m - x \in P_i$ and $2m - x \ne x$;
3. for all $x, y \in P_i$ with $x \ne y$: $c(|x - y|) \ne i$ and $c(|x - y|) \ne q$.

In short,

So $P_i$ is the union of $u/2$ pairs $\{m - d, m + d\}$, the reflection $x \mapsto 2m - x$ preserves all differences inside $P_i$, and the difference colouring on $P_i$ uses at most $k$ of the $k + 2$ colours. For six colours under $R_4(3) \le 61$ ($k = 4$, $u = 60$, $m = 900$), each of the five sets $P_i$, $i \ne c(901)$, consists of $30$ pairs $\{900 - d, 900 + d\}$, and its differences use only the four colours other than $i$ and $c(901)$.

Formalization Note. The colouring is a function on all of $\mathbb{N}$, and SchurColoring (2 * m + 1) c constrains only its values on $[1, 2m+1]$; the case $x = y$ is included, so $c(2x) \ne c(x)$ for $1 \le x \le m$. In TriangleRamsey k (u + 1) the colours are natural numbers from any Finset of at most $k$ elements, and the pair colouring is a function $\mathbb{N} \to \mathbb{N} \to \mathbb{N}$ that is constrained only on the pairs $x < y$ of the vertex set; for $k \ge 1$ the statement is $R_k(3) \le u + 1$. For $k = 0$ the hypotheses force $u \ge 2$ and $m \ge 2$, and $[1, 2m+1] \supseteq [1, 5]$ has no Schur colouring with two colours, so the statement is vacuous there; the case $k = 1$ is not vacuous. $P_i$ is the Finset endpointNbhd c m i $= \Gamma_i(V_m, 2m + 1)$. The subtraction $2m - x$ is truncated, but every $x \in P_i$ satisfies $x \le 2m$ (it lies in $[0, 2m+1]$ and differs from $2m + 1$), so $2m - x$ is the ordinary difference. Distances are Nat.dist. The point $0$ can lie in $P_i$, and then $2m$ does too. The statement is vacuous for $u = 0$, since $\mathrm{TR}(k, 1)$ is false. Example ($k = 1$, $u = 2$, $m = 6$): for the Schur colouring of $[1, 13]$ with classes $\{1, 4, 7, 10, 13\}$, $\{2, 3, 11, 12\}$, $\{5, 6, 8, 9\}$, the two endpoint neighbourhoods are $\{2, 10\}$ and $\{5, 7\}$; both are closed under $x \mapsto 12 - x$, and their differences $8$ and $2$ have the colours of $\{5, 6, 8, 9\}$ and $\{2, 3, 11, 12\}$.