Problems
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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)…
Does every globally asymptotically stable homogeneous polynomial vector field admit a homogeneous polynomial Lyapunov function? No. A planar homogeneous cubic vector field with integer coefficients is globally asymptotically stable yet…
Can one hear the shape of a drum, in the Steklov setting and in the plane? No: there exist pairs of noncongruent bounded plane domains with identical Steklov spectra including multiplicities, simply connected, strictly convex, with…
Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A 14-vertex witness has signature 2, and chaining copies gives connected line graphs of signature k + 1 for…
Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions…
A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold K(π,1) spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Curvature is expected to smooth flame-front wrinkles and so reduce turbulent flame speed, and in two-dimensional shear flows this was proved. In three dimensions it fails: there is a smooth periodic shear flow for which introducing…
Is the sequence W_0, W_1, …, W_n counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no…
Mason conjectured the following: let M be a matroid of rank r, and let W_i denote the number of flats of M of rank i. Is it true that for all 1 ≤ i ≤ r - 1, we have W_i^2 ≥ W_i + 1W_i - 1? This is false; a counterexample is given by a…
Is every weakly quasi-complete Noetherian local ring quasi-complete? Asked by D. D. Anderson in 2014. The ring A = k^p[[X, Y]][k] with k = F_p(u_1, u_2, …) is weakly quasi-complete but not quasi-complete.
An n-divisor set contains a multiple of every integer from 1 to n. Umans and Wang proposed, as the arithmetic-progression form of their Strong (α,β)-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude,…
The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal…
The Gaussian Moments Conjecture asks whether, for complex polynomials P,Q in n independent standard real Gaussian variables, E(P^m)=0 for all m≥ 1 forces E(QP^m)=0 for all large m. Explicit counterexamples with E(P^m)=0 and E(QP^m)=m!≠ 0…
Erdos and Szemeredi conjectured that every finite set of reals satisfies max(|A+A|,|AA|) ≥ |A|^2-o(1). False: there are arbitrarily large A ⊂ R, of algebraic integers in a number field of degree asymp log|A|, with max(|A+A|,|AA|) ≤ |A|^2-c…
Do a finite group's order together with ∑_g ∈ G φ(|g|) determine whether the group is simple? A simple and a non-simple group of order 6048 share the statistic 23984.