Problems

For every connected graph, is the deviation of its adjacency eigenvalues at most its order divided by its average distance? Exact lollipop-graph certificates refute the inequality when deviation means population standard deviation, under…

An ℓ-Oddtown is a family of subsets of an n-element set whose set sizes are not divisible by ℓ while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is n for prime ℓ, Babai and Frankl extended this to…

A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different.…

Godara and Sarkar proved d(H_27)=6 for the exponent-p Heisenberg group and posed d(H_p^3)=3p-3 for every odd prime p, leaving p≥5 open. The paper settles the first open case, d(H_125)=12, the upper bound reducing to a finite spread bound…

The paper constructs an exact cluster F⊆Z^2 of cardinality 8 with full affine span and an F-tiling whose orbit closure contains no 1-periodic F-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows.…

The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network are exactly its target-free cliques, the bidirected cliques no outside vertex receives an edge…

R(c) = 40c+41 for every c ≥ 2 such that c+1 is divisible by 3, 4, 5, or 7 (covering ≈ 66% of all c); the full conjecture (Myers 2015 Conj. 4.9, ABEMRS16 §5.5) reduces to prime cases p ≥ 89, all smaller primes settled by SAT. Twenty-eight…

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…

Espuny Diaz, Lichev and Wesolek conjectured that a Dirac-type minimum degree condition forces Hamiltonicity in spanning subgraphs of cycle powers. Asymptotically true: for every ε > 0 and all large k, any spanning subgraph of the kth power…

Let H(n) be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than n. With k_1 = 1 and k_n = ⌊ n/2 ⌋ + k_⌊ n/2 ⌋ + k_⌈ n/2 ⌉, prove H(n) ≥ c k_n for some constant c > 1, already for n…

Are the Kazhdan-Lusztig polynomials of matroids always unimodal - in particular log-concave, or even real-rooted, as conjectured? No: representable matroids obtained by deleting points from finite projective geometries have non-unimodal…

A t-(v,k,λ) covering is a family of k-subsets of a v-set meeting every t-subset at least λ times, and C(v,k,t) is the least number of blocks. The recorded bounds for C(12,6,4) were 40 ≤ C(12,6,4) ≤ 41. No 4-(12,6,1) covering with 40 blocks…

Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A 14-vertex witness has signature 2, and chaining copies gives connected line graphs of signature k + 1 for…

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

Escobar, Klein and Weigandt proved that gradedness of an ASM weak order interval, constancy of Coxeter length across its fibres, and equidimensionality of the associated ASM varieties are mutually equivalent, and conjectured (Conjecture…

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…

Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions…

The imbalance of an edge uv of a finite simple graph is the absolute difference of the degrees of u and v. Kozerenko and Skochko conjectured that the multiset of all edge imbalances is graphic - realizable as the degree sequence of some…

A generalization of Boppana's entropy inequality, of the kind used in union-closed-sets arguments, proved and formalized: the sharp form with the extremal constant characterized via the unique positive solution of an explicit equation.

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…