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“Computations in some cases with n = 3 suggest that if S is a generating set for Γ0, then F1 = F0 and ⟨S⟩Γ is precisely the collection of points L of Γ for which there exists an i-element E of Δ(F0) such that L ⊂ E^⟂ (in…”

“If G is a graph where every precoloring of at most k edges can be extended to a proper χ'(G)-edge coloring, then every precoloring of at most k+1 edges of G square K_2 is extendable to a proper (χ'(G)+1)-edge coloring…”

“For some fixed n, the POLYTOPE TRANSLATION problem for rational simplices Δ ⊂R^n is NP-hard.”

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?

Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and…

For a pure O-sequence h = (h_0, …, h_e) of codimension three and type two, is h_i^2 ≥ h_i-1 h_i+1 for every interior index i? The stated monomial case is proved; the broader level-Hilbert-function case remains open.

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.

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

Given n and 1 ≤ c ≤ n!, can n distinct group elements be chosen so that their n! ordered products take exactly c distinct values? Constructions realize every c.

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

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?

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.

The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have…

Does there exist A=a_1<a_2<…⊂ N which is a minimal basis of order 2 (every large integer is the sum of 2 elements from A, and no proper subset of A has this property) such that lim_k→ ∞a_k/k^2=c for some c≠ 0? A claimed construction gives…

If A(x) counts integers satisfying the Sylow divisor condition, determine the constant c in A(x)/x = exp(-(c + o(1)) √log x loglog x). The claimed exact value is c = 1/(2√log 2).

For |A| = n, how small can the cofactor set Q(A) = a / gcd(a,b) : a, b ∈ A be? The answer is h(n) = n^1/2 + o(1): a new upper bound h(n) ≤ n^1/2 exp(O(√log n)) matches the classical lower bound.

What is the maximum volume of a convex body in R^n whose centroid is its only interior lattice point? Ehrhart conjectured the extremal value in 1964; the sharp maximum is now determined in every dimension.

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.

Bounds the weighted sum ∑ 1/(a log a) taken over primitive sets of integers (sets where no element divides another).

Is every group sofic - does every group admit approximate finite permutation representations? A central open question of geometric group theory since Gromov introduced soficity: soficity implies Gottschalk's surjunctivity conjecture,…

Does there exist a group with more than one but only finitely many maximal locally soluble normal subgroups? An explicit group with exactly two settles it.

For every finite set A⊂ Z with |A|≥ 2, define C(A)=log(|A+A|/|A|)/log(|A-A|/|A|). Determine the largest possible value of C(A), equivalently the least universal exponent c such that |A+A|/|A| ≤ (|A-A|/|A|)^c for every such set A. The…