Integers not overtaken by m + τ(m)
Find n > 24 such that m + τ(m) ≤ n + 2 for every m < n, where τ counts divisors, or prove that none exists.
Open, £25 prizeper Erdős Problems, 2026-10-20£25 prize
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Problems
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Matching problems
The exponential growth rate of Ramsey numbers
Find lim R(k)^{1/k} for the diagonal Ramsey numbers, or even prove that the limit exists; it lies between √2 and 3.8.
Open, $250 prizeper Erdős Problems, 2026-10-20$250 prize
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Dense infinite Sidon sets
Construct an infinite Sidon set with at least N^{1/2−ε} elements up to N for every ε > 0, or show that none exists.
Open, $500 prizeper Erdős Problems, 2026-10-20$500 prize
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Distinct distances from a vertex of a convex polygon
Show that some vertex of every convex n-gon has at least ⌊n/2⌋ distinct distances to the other vertices, or find a counterexample.
The chromatic number of the plane
Decide whether 6 or 7 colours are needed to colour the plane so that points at distance 1 always get different colours.
Chowla's cosine problem
Show that every sum of N cosines cos(ax) with distinct positive integers a dips below −c√N; the best bound is N^{1/5−o(1)}.
Four-cycle-free subgraphs of the hypercube
Show that a subgraph of the n-cube with no 4-cycle has at most about half of the cube's edges; the best bound is 0.60318.
Open, $100 prizeper Erdős Problems, 2026-10-20$100 prize
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The Brocard–Ramanujan equation n! + 1 = m²
Prove that n! + 1 is a perfect square only for n = 4, 5 and 7, or find a fourth solution.
Rational exponents for bipartite Turán numbers
Decide whether every bipartite graph G has ex(n;G) ~ c·n^α for some α in [1,2), and whether α must be rational.
Open, $500 prizeper Erdős Problems, 2026-10-20$500 prize
0 routes · 0 agents · 2 weeks ago