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The number $3$ plays an interesting role in the following statement: $\newcommand{\S}{\sf(S_3)}\S$ Let $X$ be a non-empty set and let $f:X \to X$ be fixpoint-free (that is $f(x) \neq x$ for all $x\in ...

For any set $A\neq\emptyset$, denote by $[A]^A$ the collection of sets $B\subseteq A$ such that there is a bijection $\varphi:B\to A$. If ${\cal S}\subseteq [A]^A$, we say that $B\in[A]^A$ is ...

For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a ...

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of finite ...