Compactness theory for vicinity-based version of Ramsey theory A set of lecture notes has claimed that: > The infinite version of Ramsey theorem implies the finite version by means of compactness theorem: $\forall k \in \mathbb{N}, \exists > n\in\mathbb{N}$ such that every finite graph with at least $n$ vertices contains a clique of size $k$ or an independent set of size $k$. I've found Brian's answer here, where the relation between infinite and finite versions by compactness theorem is explained based on the coloring derivation of Ramsey theorem. I'm wondering how one can define the $\Sigma$ set of sentences (and its extension to be used by compactness theorem) for the aforesaid claim based on vicinity notions like clique and independent?

If I understand you correctly, you're asking how to go from colorings to cliques/independences. These are actually the same thing:

* Given a graph $G$ on a set $X$, we get a 2-coloring $C_G$ of pairs of elements of $X$ given by $C_G(\\{a, b\\})=1$ iff there is an edge between $a$ and $b$ in $G$.

* Given a 2-coloring $C$ of pairs of elements of $X$, we get a graph $G_C$ on $X$ by placing an edge between $a$ and $b$ exactly when $a\
ot=b$ and $C(\\{a, b\\})=1$.




This gives you a method for translating everything from one language to another. For example, if we equate graphs/colorings in this way, then a clique is exactly a homogeneous set for the color $1$, and an independent set is a homogeneous set for the color $0$.