• Problem 1026 on the Erdős problem web site recently got solved through an interesting combination of existing literature, online collaboration, and AI tools. • The purpose of this blog post is to try to tell the story of this collaboration, and also to supply a complete proof. • The original problem of Erdős, posed in 1975, is rather ambiguous. • Erdős starts by recalling his famous theorem with Szekeres that says that given a sequence of distinct real numbers, one can find a subsequence of length which is either increasing or decreasing; and that one cannot improve the to , by considering for instance a sequence of blocks of length , with the numbers in each block decreasing, but the blocks themselves increasing. • He also noted a result of Hanani that every sequence of length can be decomposed into the union of monotone sequences. • He then wrote “As far as I know the following question is not yet settled.
Article Summaries:
- In 2026 the long‑standing Erdős problem #1026, originally posed in 1975 and ambiguously framed on the Erdős problem website in 2025, was finally resolved. The solution emerged from a blend of existing combinatorial results, a coordinated online collaboration, and AI‑assisted computation. Researchers clarified the problem’s intent, reformulating it in terms of a game where Alice partitions coins into piles and Bob selects a monotone subsequence. The team proved the optimal constant governing the maximum fraction Bob can guarantee, thereby completing the proof and answering Erdős’s question.
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