Problems
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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 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…
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…
Improved lower bounds for nine classical Ramsey numbers, including R(3,13) ≥ 61, R(3,18) ≥ 100, and seven R(4,k) records up to R(4,20) ≥ 237, found by AlphaEvolve-discovered search algorithms.
Is the sequence W_0, W_1, …, W_n counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
Mason conjectured the following: let M be a matroid of rank r, and let W_i denote the number of flats of M of rank i. Is it true that for all 1 ≤ i ≤ r - 1, we have W_i^2 ≥ W_i + 1W_i - 1? This is false; a counterexample is given by a…
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.
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…
How large can a Bruhat interval in S_n that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension O(n log n) for n a power of 2, matching the largest possible…
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…
Erdos and Szemeredi conjectured that every finite set of reals satisfies max(|A+A|,|AA|) ≥ |A|^2-o(1). False: there are arbitrarily large A ⊂ R, of algebraic integers in a number field of degree asymp log|A|, with max(|A+A|,|AA|) ≤ |A|^2-c…
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…
For A ⊂ F_p of density 1/2, call A almost affine invariant under φ(x) = ax+b if |A triangle φ(A)| = o(p). Problem 90 asks for the threshold K below which A can be almost affine invariant simultaneously under all such φ with |a|, |b| ≤ K…
Tuza conjectured that every finite simple graph satisfies τ(G) ≤ 2ν(G), where ν counts pairwise edge-disjoint triangles and τ is the fewest edges whose deletion leaves the graph triangle-free. Puleo had proved it for maximum average degree…
Let T_k be the least t such that every equinumerous t-coloring of [tn] contains a rainbow k-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured T_k = Θ(k^2); Conlon, Fox and Sudakov proved T_k = O(k^2 log…
Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial D(G, x), proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a…
Seymour conjectured that every oriented graph has a vertex x with |N^++(x)| ≥ |N^+(x)|. It holds for oriented graphs of minimum out-degree exactly 7, the first improvement to the out-degree threshold since Kaneko and Locke settled degree 6…
The interchange graph G(R,S) has the (0,1)-matrices with row sums R and column sums S as vertices, adjacent when they differ by a single 2× 2 interchange. Brualdi asked whether G(R,S) is always Hamiltonian. It satisfies more: it is…