Erdős problem 138 · problems.science

If $W(k)$ is the least $N$ such that every two-colouring of $\{1, \dots, N\}$ contains a monochromatic $k$ -term arithmetic progression, must $W(k+1) - W(k) \to \infty$ ?

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FormalConjectures/ErdosProblems/

138.lean

Retained formal statement7 of 9

Asserts that for any number of colors r and any progression length k, there always exists some number N large enough to guarantee a monochromatic arithmetic progression. In other words, the set monoAP_guarantee_set is non-empty. This is the fundamental existence result that allows for the definition of the van der Waerden numbers.

FormalConjectures/ErdosProblems/138.leanErdos138.monoAP_guarantee_set_nonempty1 lineExact file

1∀ (r k : ℕ), (Erdos138.monoAP_guarantee_set r k).Nonempty

SolvedStatement only, no proof

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