Concept

Compactness

The property that a statement about an infinite object follows from the same statement about all its finite parts. It turns finite combinatorial theorems into infinite ones and back, and is what makes a colouring of every finite stretch of the integers extend to a colouring of all of them.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A tree branching at most 3 ways, to depth 4, and the path through it. A tree drawn level by level, with the nodes that die out faint and a highlighted path that always steps to a node with descendants at the bottom.

An infinite tree has an infinite path

A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.

logic · models

Two colours avoid a progression up to 8, and no further. The numbers 1 to 8 in the two colours that avoid three equally spaced numbers in one colour, with the number 9 beside them in both colours and the pattern each choice forces.

Three in a row on the number line

Colour the numbers one to eight in two colours and it can be arranged that no three equally spaced numbers agree. Add the ninth and it cannot. The structure being forced is arithmetic rather than graphical, and the proof is a different proof.

discrete · ramsey theory

Named alongside it

The objects these essays reach for when they reach for this one.

Arithmetic progression Counting argument Exhaustive search Existence proof Fermats little theorem Finite branching Graph colouring Infinite path Konig lemma Modular arithmetic Pigeonhole Ramsey number

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