Explore the archive

For every connected graph, is the variance of its positive adjacency eigenvalues at most its order divided by its average distance? Exact dumbbell-graph certificates refute the bound under both conventions for average distance.

For S(x) = #(a,b) : a + b ≤ x, σ(a) + σ(b) = σ(a+b), is S(x) ~ cx? The preprint claims S(x) grows faster than x (log x)^R for every fixed R, ruling out the linear asymptotic.

For a differential poset P, must the weighted 2-multichain series M_P,2(q) be a rational multiple of F_P(q)^2, the square of its rank generating series?

What is the shortest curve guaranteed to reach the boundary of the golden gnomon - the isosceles triangle with equal sides 1 and apex angle 108^∘ - from an unknown starting position and heading? The optimum is a symmetric seven-piece path…

For every zonotope Z ⊂ R^d and vectors v_1,…,v_n ∈ Z, there are signs with ∑_i x_i v_i ∈ C√d Z for a universal constant C. This resolves a 2002 conjecture on vector balancing in zonotopes.

The Courtade-Kumar conjecture (2014) posits that dictatorship functions maximize mutual information between a Boolean function's output and a noisy input. The paper resolves an open question posed by Courtade and Kumar themselves - a sharp…

Is the density of Chernoff's distribution - the law of argmax_t W(t) - t^2 for two-sided Brownian motion W - strongly log-concave, as conjectured by Balabdaoui and Wellner in 2014?

Is the maximum size of a set A⊆ 1,…,N such that ab+1 is never squarefree (for all a,b∈ A) achieved by taking those n≡ 7pmod25? Resolved for all sufficiently large N: any near-maximal A is contained in n≡ 7pmod25 or n≡ 18pmod25, leaving…

At the critical inverse temperature β=1 in the Sherrington-Kirkpatrick spin glass model, Talagrand conjectured that the expected squared overlap of two independent Gibbs replicas has an exact N^-2/3 scaling: there exists a constant a>0…

Frankl, Peng, Rodl and Talbot asked in 2007 whether the set of Turan densities of families of r-graphs contains intervals. It does: for every r ≥ 3 the set contains non-degenerate intervals, including one of the form [1-δ_r, 1].

Let F(n) be the largest A⊆1,…,n with anmid bc for distinct a,b,c∈ A. Is F(n)=π(n)+(C+o(1)) n^2/3(log n)^-2 for some constant C?

Is the zero forcing number of every connected graph with maximum degree 3 at most its independence number plus one? A connected 24-vertex subcubic graph with independence number 9 and zero forcing number 11 refutes this 2017 TxGraffiti…

For a closed infinite set F ⊆ C, let μ(F) be the infimum of |z : |f(z)| < 1| over monic polynomials with zeros in F. Is μ(F) determined only by the transfinite diameter of F?

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

After L^2 normalization, stable phase retrieval holds over the L^2-spans of independent real-valued centered random variables exactly when all but possibly one coordinate satisfies a uniform two-sided L^1 bound. This confirms the…

Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a…

For a single-source unsplittable flow, find the optimal universal additive constant C s.t. every feasible fractional flow x with arc costs c should admit an unsplittable routing y with c^top y ≤ c^top x and y_a ≤ x_a + C · D on every arc.…

Klopp and Zadik gave an exponential-time node-private algorithm for exact community recovery in stochastic block models and asked whether a polynomial-time algorithm could match it. One can: a Lipschitz surrogate for the penalized…

The Lonely Runner Conjecture of Wills and Cusick states that among k+1 runners at distinct constant speeds on a unit circle, each runner is at some time at distance at least 1/(k+1) from all others. Following Rosenfeld's computer-assisted…

Han's conjecture predicts that a finite-dimensional algebra with eventually vanishing Hochschild homology has finite global dimension. Within the tau-Hochschild framework it splits into persistence and survival, and the Liu-Morin extension…