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Let X be a set of cardinality ℵ_ω and f a function from the finite subsets of X to X such that f(A)not∈ A for all A. Must there exist an infinite independent Y⊆ X, i.e. with f(B)not∈ Y for all finite B⊂ Y? Claimed resolution: the positive…

What is the largest possible measure of a subset of a radius-R disk in R^2 containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives M(R) ≪ R^1/2; with Sárközy's lower construction, M(R) = R^1/2 +…

Does every nontrivial finite simple graph have noninteger Sombor energy? If ρ_1,…,ρ_n are the eigenvalues of the Sombor matrix of a graph G, its Sombor energy is E_SO(G)=∑_i=1^n|ρ_i|. The conjecture asserted that E_SO(G)∉ Z for every…

Let k≥ 3 and f_k(N) be the maximum of ∑_n∈ A1/n over all A⊆1,…,N containing no k subsets with the same pairwise least common multiple. Estimate f_k(N). The claimed answer: f_k(N)=(log N)^γ_k+o(1), where γ_k is a weighted generalization of…

Is there an entire non-zero function f:C→ C such that, for any infinite sequence n_1<n_2<…, the set z: f^(n_k)(z)=0 for some k≥ 1 is everywhere dense? The literal question is trivial for polynomials, so the claims address the…

For a connected graph G , let t= tree( G ) (order of a largest induced tree), A= average eccentricity, and L= maximum independence number of a neighbourhood. Then ⌈ (A+L)/3 ⌉ ≤ t. (The evenly-divided reading of the conjecture holds; a…

Let n_1<n_2<… be a lacunary sequence of integers and f∈ L^2([0,1]) with nth Fourier partial sum f_n. Is there an absolute constant C>0 such that if | f-f_n|_2 ≪ (logloglog n)^-C then 1/N∑_k≤ Nf(α n_k)→∫_0^1 f for almost every α? A preprint…

Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below (1+ε)n to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.

The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly t red edges. Introduced by Papadimitriou and Yannakakis in 1982, it has been in randomized polynomial time…

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.

Let k≥ 3 and A be an additive basis of order k. Does there exist a constant c=c(k)>0 such that if r(n)≥ clog n for all large n (where r(n) counts representations of n as a sum of at most k elements of A) then A must contain a minimal basis…

In the square Gaussian binary MIMO model y = √ρ/N Hx^⋆ + w, exhaustive maximum-likelihood detection recovers x^⋆ once ρ > 2log N, while sphere decoding at that threshold scale costs expΘ(N/log N). Whether any polynomial-time detector…

Let L(x^ay^b)=a! b! on C[x,y]. The Factorial Conjecture asks whether L(f^m)=0 for every m≥ 1 forces f=0. The homogeneous two-variable case was settled by Liu and Sun; the inhomogeneous problem does not reduce to it, because radial…