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A finite closure system can be given by implications or by a list of subsets closed under intersection. Deciding whether one specification of each kind defines the same family had remained open in several settings; the paper proves the…

For the switch-walk-switch walk on Z_2 wr Z started at (0,0) and (0,2), prove |P_t^x - P_t^y|_TV asymp t^-1/2.

A collection of open problems from the algebraic and enumerative combinatorics literature, resolved in one paper: a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas and Williams on the echelonmotion operator on modular…

R_dih(P_a^alt, K_b) = 1 + (a-1)(b-1) for all a ≥ 4, b ≥ 1 — the a ≥ 4 slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the a = 3 case (see sibling entry), this resolves Conjecture 4.9 in full for a ≥ 3.

Fulek defined a weight-five three-row 0-1 matrix L_3 and asked whether ex(n, L_3) = O(n). It is: every r × s matrix avoiding L_3 has at most 27r + 2s ones, so 6n - 8 ≤ ex(n,L_3) ≤ 29n for n ≥ 5. The same argument covers an infinite family…

Can S-decoding polynomials modulo a product of k primes be built with only k+1 nonzero coefficients, the minimum their own lower bound allows? Yes, via a general framework for special prime products. The consequence is that for any…

Does quantum memory give a query-complexity advantage for learning an unknown quantum channel, when protocols without it must measure after each channel use and keep only a classical transcript? It does, and the paper also determines how…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet, settled in the ternary case. For independent X_1, …, X_n taking values in 0,1,2, the entropy of S_n = X_1 + ……

R_dih(P_3^alt, K_b) = R_cyc(P_3^alt, K_b) = 2b - 1 for all b ∈ N — the a = 3 slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817) and Conjecture 4.23 (Bašić–Damnjanović–Stevanović–Stošić, arXiv:2604.16188).

The directed five-dimensional torus D_5(m) has a Hamilton decomposition for every odd m ≥ 3, extending the decomposition program for directed tori beyond the three-dimensional case.

Nikolov and Ullman asked, as Open Problem 1 on DifferentialPrivacy.org, whether k statistical queries over a universe of size T can be released under pure differential privacy at the square-root error rate that the known lower bounds…

Let S(N) count the distinct values of ∑_n∈ A 1/n over A⊆1,…,N. Estimate S(N).

Monical, Tokcan and Yong conjectured that every fixed positive power of the Vandermonde determinant fails to have saturated Newton polytope in sufficiently many variables. For every even power k ≥ 4 there is an explicit lattice point of…

There exists a Hadamard matrix of order 668: a matrix H∈-1,1^668×668 such that HH^ T=668I_668. Equivalently, the 668 rows of H are pairwise orthogonal.

Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers f(n) = B_n(-1) take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result…

Let n_k be the least n > 2k such that (n-k)(n-k+1)…(n-1) has no prime factor in (k, 2k). Erdos conjectured a superpolynomial lower bound; for all large k, n_k > e^log^2 k / (20 loglog k).

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

A collection of open problems drawn from published lists, including Cahen, Fontana, Frisch and Glaz's Open Problems in Commutative Ring Theory and Erman and Sam's survey of Boij-Soderberg theory, each proved or disproved by one automated…

A problem from Fajtlowicz's Graffiti program, studied by Erdős and Staton, on the Havel-Hakimi residue of common-divisor graphs. The paper resolves the problem and extends it, determining the residue's first-order scale and its nontrivial…