Is Ramsey’s theorem the pigeonhole principle on stilts?
PAPER II — MATHEMATICS · QUESTION 35 OF 35
In plain terms: The free founding paper in which counting is built up into order — the stilts assembled, step by step.
The question as set
Is Ramsey’s theorem the pigeonhole principle on stilts? Exhibit the principle as the theorem’s one-dimensional case, derive the finite two-colour theorem from it by iteration, and then let the calibrations judge: say what the growth of the Ramsey numbers and the reverse-mathematical strength of the theorem for pairs each establish that counting alone does not.
The question in Paper II · Its section on the reference page
The question in context
This is question 35 of the 35 set in Paper II — Mathematics of The Truly Hardest Exam in the World, the examination of The College of All Minds. In the College’s subject index it is filed under Foundations of mathematics. Like every question of the College, it carries three references of record, verified live at publication and described twice — technically, and in plain terms — so the ground can be judged before it is walked. The response required is an argument, not a survey, of at most three thousand words, as technical as the question demands and no more.
The references of record
A combinatorial problem in geometry
The derivation the question demands, performed at the origin: iterated pigeonhole compounded into Ramsey-type bounds, with the monotone-subsequence lemma as the principle in miniature.
In plain terms: The free founding paper in which counting is built up into order — the stilts assembled, step by step.
An exponential improvement for diagonal Ramsey
The growth calibration: the first exponential improvement of the diagonal upper bound since 1935, placing R(k) below (4−ε)^k — the measured distance between the theorem and its one-dimensional case.
In plain terms: The free breakthrough paper showing how hard the numbers push back — the gap between the dovecote and the party, made quantitative.
On the strength of Ramsey’s theorem for pairs
The logical calibration: the theorem for pairs located strictly above the infinite pigeonhole principle in the reverse-mathematical hierarchy — the adjudication delivered as a theorem.
In plain terms: The free landmark paper proving, in the exact currency of logical strength, that Ramsey for pairs is more than pigeonhole — the question’s verdict, formalised.
Background: Pigeonhole principle · Ramsey’s theorem