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1011 problems

Open problems

Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.

All fields Mathematics 936Computer science 28Physics & astronomy 22Chemistry & materials 13Life sciences 6Earth & society 6
Combinatorics 158Number theory 500Graph theory 71Algebra 45Geometry 72Analysis 69Probability 11Logic & formalisation 10
Source Any Curated here 17Formal Conjectures (Lean) 116Optimization constants 17AlphaEvolve problems 8
Verification AnyA · Machine-checkableB · ReproducibleC · Reviewed Tier AnyStandardHardGrand challenges

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A Hard Combinatorics · Formal Conjectures (Lean)

Ramsey numbers

The open problem: determine the Ramsey number R(5,5). It is known that 43 ≤ R(5,5) ≤ 46.

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A Hard Combinatorics · Formal Conjectures (Lean)

Sparse Ruler

Wichmann's conjecture on optimal rulers. Every optimal ruler of sufficiently large length is a Wichmann ruler W(r, s) (up to reflection, i.e. reversing the segment list). Posed by Wichmann [Wi63].

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A Hard Combinatorics · Formal Conjectures (Lean)

Steiner Systems

Construct an S(t, k, n)-Steiner system with n > k > t > 5, t < 10, and n < 200. No example of a Steiner system with t > 5 is known, despite a 2014 existence theorem by Keevash showing that such systems must exist for sufficiently large n. Reference: Large Steiner Systems

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A Hard Combinatorics · Formal Conjectures (Lean)

Written on the Wall II - Conjecture 40

WOWII Conjecture 40 For a nontrivial connected graph G the size f(G) of a largest induced forest satisfies f(G) ≥ ceil((p(G) + b(G) + 1)/2) where p(G) is the path cover number and b(G) is the largest induced bipartite subgraph size.

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