Consider the following statements:
Finite Ramsey Theorem ($\mathrm{FRT}$): For any $k,r,p\in\mathbb{N}^{+}$, there exists $N\in\mathbb{N}$ such that every function $c:[N]^{p}\rightarrow [r]$ has a homogeneous subset of size $k$.
By $[r]$ we mean the set $\{1,\ldots r\}$, and by $[A]^{p}$ we mean the set of $p$-element subsets of $A$. If $A=[N]$ we simply denote $[[N]]^{p}$ by $[N]^{p}$. A subset $H\subseteq A$ is homogeneous for $c$ if $c|_{[H]^{p}}$ is constant. Finally, the function $c:[A]^{p}\rightarrow [r]$ is called an $r$-coloring of $[A]^{p}$
Infinite Ramsey Theorem ($\mathrm{IRT}$): For any $r,p\in\mathbb{N}^{+}$, any $r$-coloring $c$ of $[\mathbb{N}]^{p}$ has an infinite homogeneous subset.
I am trying to prove that $\mathrm{IRT}\Rightarrow\mathrm{FRT}$ via the Compactness Theorem for propositional logic.
The idea is to conclude that $\mathrm{IRT}$ fails if $\mathrm{FRT}$ fails for some $k$. Indeed, assume $\neg\mathrm{FRT}$. That is, for any arbitrarily large $N\in\mathbb{N}$ there is an $r$-coloring $c_{N}$ of $[N]^{p}$ having no homogeneous subset of size $k$. Define a propositional language $L$ whose atoms are $C^{i}_{a_{1}\ldots a_{p}}$, for all $a_{1},\ldots,a_{p}\in\mathbb{N}$ and $1\leq i\leq r$ (intuitively $C^{i}_{a_{1}\ldots a_{p}}$ means $\{a_{1},\ldots,a_{p}\}$ will be $i$-colored when true). Then we define a set $S_{k}$ to be the set of $L$-sentences:
- $C^{i}_{a_{1}\ldots a_{p}}\Leftrightarrow C^{i}_{a_{f(1)}\ldots a_{f(p)}}$ for all $a_{1},\ldots,a_{p}\in\mathbb{N}$, any permutation $f$ of $\{1,\ldots,p\}$, and $1\leq i\leq r$.
- $\displaystyle\bigvee_{1\leq a_{1}<\cdots<a_{p}\leq k}\neg C^{i}_{a_{1}\ldots a_{p}}$ for all $a_{1},\ldots,a_{p}\in\mathbb{N}$, and $1\leq i\leq r$ (every $k$-element subset of $\mathbb{N}$ has at least one $p$-element subset which will not $i$-colored).
- $\displaystyle\bigvee_{1\leq a_{1}<\cdots<a_{p}\leq k}C^{i}_{a_{1}\ldots a_{p}}$ for all $a_{1},\ldots,a_{p}\in\mathbb{N}$, and $1\leq i\leq r$ (every $k$-element subset of $\mathbb{N}$ has at least one $p$-element subset which will be $i$-colored).
With 3 and 4 together I am trying to state that there will not be a homogeneous $k$-element subset for the $r$-coloring of $[\mathbb{N}]^{p}$.
Did I define the language $L$ and the set $S_{k}$ appropriately?
If $\mathrm{FRT}$ fails for $k$, then $S_{k}$ is satisfiable. Indeed, suppose that $\Delta$ is a finite subset of $S_{k}$. Then the elements $a_{i}$ mentioned in $\Delta$ form a finite set. As a consequence, there is a largest $m\in\mathbb{N}$ for which $a_{m}$ occurs in one of the finitely many atoms of $\Delta$. Since the $\mathrm{FRT}$ fails for $k$, there is $N\geq m+1$ such that $c_{N}:[N]^{p}\rightarrow [r]$ has no homogeneous subset of size $k$. In fact, WLOG we can say $[N]=\{0,\ldots,m\}\cup\{m+1,m+2,\ldots\}$. Therefore, we can define a truth assignement $$\mu(C^{i}a_{1}\ldots a_{p})=\mathsf{T}\text{, for $a_{1},\ldots,a_{p}\leq m$ iff $c_{N}(\{a_{1},\ldots a_{p}\}) = i$}.$$
I then try to show that $\mu$ must satisfy $\Delta$ by showing it satisfies formulas 1-3, but I run into trouble. What is wrong with my definition of $\mu$?
Once I can show that $\Delta$ is satisfiable, the compactness theorem implies that $S_{k}$ is satisfiable. But this means there is an $r$-coloring $c$ of $[\mathbb{N}]^{p}$ having no homogeneous subset of size $k$. Since $\mathrm{IRT}$ states that every $r$-coloring of $[\mathbb{N}]^{p}$ must have an infinite homogeneous subset, it must have a finite homogeneous subset of every size. Therefore, $\neg\mathrm{FRT}\Rightarrow\neg\mathrm{IRT}$, as desired.
This question is related.