Problems

Araujo, Piga and Schacht asked whether density and codegree both above 1/4 force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is p_0 = max_0 ≤ x ≤ 1minx^3, 1-x ≈ 0.3177, and below it there are dense 3-graphs…

For D_3(m) = vecC_m square vecC_m square vecC_m, can the full arc set be partitioned into three directed Hamilton cycles for every integer m ≥ 3?

For every finite connected simple graph G, is the order of the largest induced tree at least girth(G) - 1 + ecc(G, center(G)), where the last term is the eccentricity of the centre set? Answered affirmatively, with a Lean proof.

A cyclic meander induces a cyclic permutation on its 2n marked intersection points. Schwartz's conjecture on the quadratic growth of the associated meander number is resolved.

For every finite connected graph, is girth(G) + 1 at most the product of its largest induced-tree order and its second-smallest degree?

For an inclusion-free hypergraph on n vertices, a weight assignment w:[n]→[d] is isolating when a unique edge attains minimum weight. Faber and Harris conjectured that the number of isolating assignments is at least n∑_j=0^d-1 j^n-1,…

Norine conjectured that every red-blue edge-colouring of the n-dimensional hypercube Q_n in which antipodal edges get opposite colours contains a monochromatic path from some vertex to its antipode. The paper proves it, via a chain-level…

Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph G perfectly divisible if and only if χ(H) ≤ binomω(H)+12 for every induced subgraph H of G? False: the Paley graph P(17) satisfies the…

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

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) ≥…

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…

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…

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).

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…

Amdeberhan, Shareshian and Stanley showed a function from the theory of partition Eisenstein series counts alternating permutations with a given record partition, and asked whether a similar theory exists for record compositions,…

Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between 2|E|-|V| and 2|E| has period exactly 2, generalizing the middle rung of Levine's devil's staircase…

Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and…

Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for…

The near-quadratic Elekes-Ronyai expander conjecture over R predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of…