Problems

Monical, Tokcan and Yong conjectured that Schubitopes, the generalized permutahedra arising as Newton polytopes of Schubert polynomials and of Demazure characters of GL_n, are Ehrhart positive. Disproved by an explicit Schubitope whose…

For a bipartite graph without isolated vertices and with a unique minimum dominating set, does the proposed function m(n, γ) bound the number of edges whenever γ ≥ 2 and n ≥ 3γ? A 13-vertex bipartite graph with 22 edges exceeds the…

Mihail and Vazirani conjectured that the graph of every 0/1-polytope has edge expansion at least one. Disproved by a family of 0/1-polytopes whose edge expansion decreases exponentially in the dimension.

If a chromatic symmetric function is Schur positive, must every finite-variable specialization X_G(x_1, …, x_k) have a saturated Newton polytope? A 12-vertex bipartite graph realizes weights (6,6,0) and (8,2,2) but omits their midpoint…

Erdos and Hajnal asked whether h_r(G) = maxχ(H) : H ⊆ G, girth(H) ≥ r tends to infinity as χ(G) does, for every fixed r ≥ 4. It does in every fixed polynomial edge-density regime.

A subset A of the pointwise-ordered cube [0,1]^n is a k-antichain when it meets every chain in at most k points. The conjecture concerns the largest possible (n-1)-dimensional Hausdorff measure of such a set; it is settled here, following…

Kotzig conjectured that for every even n ≥ 4 the complete graph K_n decomposes into n-1 perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: K_n decomposes into n-1 perfect matchings of which…

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.…

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…

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…