Problems

Gill introduced the probabilistic automatic complexity A_P(w) of a string: the least number of states of a probabilistic finite automaton for which w is the unique most probably accepted string of its length. He asked whether A_P is…

Do arbitrarily large 4-chromatic edge-critical graphs exist with minimum degree bounded below by a positive constant times the number of vertices?

Erdos asked whether a finite unit-distance graph in the plane can have independence ratio below 1/4. One exists, built on the geometric fractional chromatic number framework of Matolcsi, Ruzsa, Varga and Zsamboki plus a carefully chosen…

Localizing Bernstein theory to prove lower bounds for the Lebesgue constants of Lagrange interpolation, with application to a problem of Erdős and Turán and to a conjectured bound from the interpolation literature.

Shokurov's global index conjecture, in the setting of foliations. Proved for foliations in dimension at most three, which also answers a question of Liu, Meng and Xie in dimension three.

Let A ⊂ N be infinite with no distinct a, b, c ∈ A such that a | (b + c) with b, c > a. Can |A ∩ [1, N]|/√N have positive lower limit? Must every such A fall below N^1-c infinitely often?

The classical Abbott-Hanson recurrence gives S(k+2) ≥ 9S(k)+4 for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted S-templates, a more flexible form of Rowley's template construction, yield S(k+2) ≥…

Let A(x) count n ≤ x such that every prime p | n has a divisor d > 1 of n with d ≡ 1 pmod p. Erdos asked whether A(x)/x = exp(-(c+o(1))√log xloglog x). It does, with c = 1/(2√log 2).

The Hessian conjecture HC_n asks whether every polynomial f with det Hess(f) ∈ C^× has a polynomial gradient inverse. It is known for n ≤ 3, false for n ≥ 5, and open exactly in dimension four, where it implies the plane Jacobian…

Regts and Sevenster conjectured that a complex-valued graph parameter f with f(∅)=1 has exponentially bounded edge-connection rank precisely when it is a mixed partition function. The paper proves it, with the numbers of even and odd…

For a finite-dimensional algebra A, finite global dimension forces HH_n(A) = 0 for all large n. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit…

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.

Erdős, Herzog and Piranian (1958) asked whether the regular n-gon maximizes the product of pairwise distances among n points of fixed diameter. After the recent discovery that it does not for even n, this paper proves the first exponential…

Does the Dinitz-Garg-Goemans cost-preserving unsplittable-flow rounding conjecture survive on acyclic planar instances with only four terminals? An explicit instance answers no: every cost-nonincreasing unsplittable routing has upper…

For A=0,1,2^d, write f(d) for the fewest proper sub-boxes covering every point exactly twice. Leader, Miličević and Tan asked whether f(d)≥ 2^d for all d, as Question 4.1 of the PatternBoost paper. The paper gives new bounds on f(d).

Let D_q(n) be the largest possible least degree of a polynomial omitted by a non-covering family of n distinct-modulus congruence classes in F_q[x]. What is its asymptotic size? The answer is D_q(n) = n/q-1 + O_q(1).

What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation widehat P(A) satisfies…

For a forbidden configuration F, the Anstee-Sali conjecture predicts that forb(m, F) is Theta(m^(X(F)-1)), where X(F) comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core…

Does the Hodge bundle Ω_g over the moduli stack of genus g ≥ 2 curves contain any nontrivial sub-bundles? Posed by Dawei Chen around 2015; the answer is no.