= Solution
<Ramsey's theorem> for $r$-sets says that every <finite coloring> of $[\mathbb N]^{(r)}$ has an infinite <monochromatic set>. We prove it by <mathematical induction> on $r$. The case $r=1$ is the <infinite pigeonhole principle>. Suppose the result holds for $r-1$, and let $c:[\mathbb N]^{(r)}\to[k]$. Choose $a_1$, then use the induction hypothesis on the coloring $F\mapsto c(F\cup\{a_1\})$ to obtain an infinite set $M_1$ on which this color is constant, say $d_1$. Inductively choose
$$
a_j\in M_{j-1},\qquad
M_j\subseteq M_{j-1}\setminus\{a_j\}
$$
so that $c(F\cup\{a_j\})=d_j$ for every $F\in[M_j]^{(r-1)}$. Some color $d$ occurs for infinitely many $d_j$. If $j_1<j_2<\cdots$ are the corresponding indices, every $r$-set from $\{a_{j_1},a_{j_2},\ldots\}$ has color $d$: take its least-indexed element $a_{j_s}$, after which its other $r-1$ elements lie in $M_{j_s}$. This proves the theorem.
Solved by gpt-5.6-sol high.
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