Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
49 problems
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For an extension G = A rtimes B of elementary abelian p-groups with a ∈ A satisfying C_B(a) = 1, must H = ⟨ a, B⟩ satisfy rank(Z(H) ∩ H') ≤ rank(B)? An explicit extension violates the bound.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For a finite abelian group G, let Φ(G) be the absolutely convex hull of the specified trilinear kernels and Φ'(G) its restriction where the third factor depends only on x_1 + x_2. Is Φ(G) = Φ'(G)? A counterexample over Z/3Z separates the…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Lassak conjectured that a reduced planar convex body of thickness Δ has area at most (π/4)Δ^2, the value for the disc. False: an explicit reduced body of thickness 1 has area 0.786215… > π/4 = 0.785398…, given by a closed-form support…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Conjectured upper bound on how many pairs among n points in the plane can be exactly one unit apart.
Must every connected graph satisfy the proposed upper bound on its independence number in terms of residue and largest induced-bipartite-subgraph order? The family overlineK_2r+1 ∨ (K_r sqcup K_r) violates it for every r ≥ 3.
Chromatic quasisymmetric functions of natural unit interval graphs were conjectured to have log-concave coefficients in the elementary basis. A connected 13-vertex example refutes it: for the Hessenberg function…
The Huneke–Wiegand Conjecture: Let R be a one-dimensional Gorenstein local domain, and let M be a finitely generated, non-zero, torsion-free R-module. If the tensor product M ⊗_R M^* is torsion-free, then M is a projective (hence free)…
For an arrangement of n hyperplanes in P^3_C with ℓ intersection lines and p intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects p - ℓ + n + 2 ≥ 0. An explicit arrangement built from…
Simon conjectured that every skeleton of a simplex is extendably shellable. False: for every d ≥ 3 there is a pure d-dimensional shellable simplicial complex that is not shelling completable.
Grothendieck asked whether every finite locally free group scheme of order n is killed by n (its n-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne…
The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network are exactly its target-free cliques, the bidirected cliques no outside vertex receives an edge…
Does the middle-third Cantor measure admit a Fourier frame, that is, a countable set of exponentials giving two-sided frame bounds on its L^2 space? No. The Cantor measure with base b admits no Fourier frame for any odd integer b > 1,…
If Sidon sets A, B ⊆ 1, …, N satisfy (A-A) ∩ (B-B) = 0, must binom|A|2 + binom|B|2 ≤ binomf(N)2 + O(1), where f(N) is the largest Sidon-set size in [N] - and can the bound be improved by a fixed proportion when |A| = |B|?
Wegner conjectured in 1965 that every finite family R of axis-parallel rectangles satisfies τ(R) ≤ 2ν(R) - 1, where τ is the minimum number of piercing points and ν the largest pairwise-disjoint subfamily. False, by an explicit…
Litvak conjectured in 2018 that for every p > 0 the quantity E[min_i ≤ n |g_i|^p], for g ~ N(0,Σ), is minimized over n × n correlation matrices by the Gram matrix of the regular simplex in R^n-1. False: the matrix Σ^cos_ij = cos(π(i-j)/n)…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.