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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 97Computer science 8Physics & astronomy 1
Combinatorics 17Number theory 24Graph theory 4Algebra 4Geometry 14Analysis 24Probability 10
Source Any Curated here 3Optimization constants 4AlphaEvolve problems 3
Verification AnyA · Machine-checkableB · ReproducibleC · Reviewed Tier AnyStandardHardGrand challenges

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B Graph theory · Optimization constants

Conway thrackle constant

In topological graph theory, a thrackle is a drawing of a finite graph in the plane in which every pair of edges meets precisely once, either at a common endpoint or at a proper crossing.

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B Graph theory · Optimization constants

Shannon capacity of the 7-cycle

Let C_7 denote the cycle graph on 7 vertices. We define C_9 to be the Shannon capacity of mathcal C_7: C_9 := Θ(mathcal C_7), where for a graph G, the Shannon capacity Θ(G) is defined by Θ(G) := sup_n ≥ 1 α(G^boxtimes n)^1/n.

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B Graph theory · Optimization constants

The coefficient of the acyclic chromatic index

Let G be a simple graph. The acyclic chromatic index χ_a'(G) of G is defined to be the least number of colors needed to color the edges of G so that no two edges coincident on the same vertex are homochromatic and there is no cycle whose edges are colored with only two colors.

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