Mathematical logic and foundations Arithmetic and incompleteness Theorem

Paris–Harrington theorem

Statement

Call a finite set $Y \subseteq \N$ large if $\abs{Y} \ge \min Y$ . The strengthened finite Ramsey theorem $\mathrm{PH}$ is the following statement. For all positive integers $n$ , $k$ , $m$ there is an integer $N$ with this property. For every colouring of the $n$ -element subsets of $\set{1, 2, \ldots, N}$ with $k$ colours, there is a large set $Y \subseteq \set{1, \ldots, N}$ with $\abs{Y} \ge m$ . Moreover, all $n$ -element subsets of $Y$ have the same colour.

$\mathrm{PH}$ is true; it follows from the infinite Ramsey theorem.
$\mathrm{PH}$ , written as a sentence of first-order arithmetic, is not provable in Peano arithmetic $\mathsf{PA}$ .

Remarks

Part 1 is a compactness argument. If $\mathrm{PH}$ failed for some $n, k, m$ , the bad colourings for larger and larger $N$ would fit together by König's lemma. They would give a colouring of the $n$ -element subsets of $\N$ whose restriction to each $\set{1, \ldots, N}$ is bad. The infinite Ramsey theorem gives an infinite homogeneous set $H$ for it. An initial segment of $H$ with $\max(m, \min H)$ elements is then large and homogeneous, a contradiction. Part 2 is proved with models. In a non-standard model of $\mathsf{PA}$ , $\mathrm{PH}$ gives initial segments that are themselves models of $\mathsf{PA}$ . From this, $\mathsf{PA}$ plus $\mathrm{PH}$ proves the consistency of $\mathsf{PA}$ . The function that sends $(n, k, m)$ to the least $N$ grows faster than every function that $\mathsf{PA}$ proves total.

Without the word "large", the statement is the ordinary finite Ramsey theorem, which is provable in $\mathsf{PA}$ .

Going deeper

For a student

Paris and Harrington proved the theorem in 1977. It was one of the first natural examples from combinatorics of a true statement independent of $\mathsf{PA}$ , after the artificial sentences of Gödel's incompleteness theorems. Goodstein's theorem is another example of the same kind.

Reading path

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Peano axioms concept
Paris–Harrington theorem statement

Tag	`509ZWA0R26`
URI	`m2://509ZWA0R26`
Kind	statement, theorem
Writing	W0 stub
MSC	03F30, 05D10
Status	This copy of the site has no build data. Run `m2 status 509ZWA0R26`.
About	Peano axioms
Uses	Peano axioms
Same as	Wikidata Q7137494 Wikipedia Paris–Harrington theorem
Changed	2 November 2026