This is an exercise I'm doing and I'd like some checking or comments.
Given a coloring $c: {\mathbb{N} \choose 3} \to C$, a set $S \subset \mathbb{N}$ is said to be
(i) rainbow if no two triples have the same colors.
(ii) c-left-injective if there is a map $c^*: \mathbb{N} \to C$ such that $c^*(i) = c(i,j,k)$ for every $i,j,k \in S$ with $i <j <k$.
(iii) c-middle-injective if there is a map $c^*: \mathbb{N} \to C$ such that $c^*(j) = c(i,j,k)$ for every $i,j,k \in S$ with $i <j <k$.
(iv) c-right-injective if there is a map $c^*: \mathbb{N} \to C$ such that $c^*(k) = c(i,j,k)$ for every $i,j,k \in S$ with $i <j <k$.
Prove or disprove the following statement.
Let $c: {\mathbb{N} \choose 3} \to C$ be a coloring. Then there exists some infinite subset $S \subset \mathbb{N}$ such that:
(1) S is monochromatic, or
(2) S is rainbow, or
(3) S is c-left-injective, or
(4) S is c-middle-injective, or
(5) S is c-right-injective
My guess is that this statement is not true, since we can come up with the following type of coloring that isn't implied by any of the previous ones:
Given a coloring $c: {\mathbb{N} \choose 3} \to C$, a set $S \subset \mathbb{N}$ is said to be
(v) c-left/middle-injective if there is a map $c^{**}: {\mathbb{N} \choose 2} \to C$ such that $c^{**}(i,j) = c(i,j,k)$ for every $i,j,k \in S$ with $i <j <k$.
c-middle/right-injective and c-right/left-injective colorings could be defined similarly.
Now assume $S$ is c-left/middle-injective. Then clearly $S$ is not monochromatic or rainbow. Let $i_1 < j_1$, $i_2 < j_2$, $i_1 \neq i_2$, and $i_1,i_2,j_1,j_2 < k$ in $\mathbb{N}$, such that $c^{**}(i_1,j_1) = c^{**}(i_2,j_2)$. Their existence is due to the infinite Ramsey theorem for $m=2$:
Theorem 1.1. For any positive integer $m$ and and $r$-colouring of $\binom{\Bbb{N}}{m}$, there exists an infinite set $S \subseteq \Bbb N$ for which $\binom{S}{m}$ is monochomatic.
Now let $c^*: \mathbb{N} \to C$ be a map such that $c^*(i) = c(i,j,k)$ for every $i,j,k \in S$, with $i <j <k$. Then we have that $c(i_1,j_1,k) = c^{**}(i_1,j_1) = c^{**}(i_2,j_2) = c(i_2,j_2,k)$, and so $c^*(i_1) = c(i_1,j_1,k) = c(i_2,j_2,k) = c^*(i_2)$. Hence $c^*$ is not injective and $S$ is not c-left-injective.
Similar reasoning shows that $S$ is not c-middle-injective or c-right-injective.
Following this pattern, I also suspect that in general $m$-uniform setting, the number of canonical colorings would be $2^m$: for any $x_1 <...<x_m$ in $\mathbb{N}$, different kinds of coloring could be defined, where $c(x_1,...,x_m)$ is determined by different combinations of elements in $\{x_1,...,x_m\}$. Please give your comments on this.