Problems
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For the Fubini numbers a(n), is a(n) = ∑_k=0^2^n-1-1 A284005(k) for every n > 0, as conjectured on the OEIS in 2018?
Every minimally generically globally rigid graph in R^d containing a subgraph isomorphic to K_d+2 is itself isomorphic to K_d+2, confirming Conjecture 6.3 of Garamvölgyi, Jackson and Jordán (2025).
WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five is at most the negative of its least distance eigenvalue. The paper refutes it with exact counterexamples of orders 38,…
Wellman and Pettie noted that the true leading constant for large-order Davenport-Schinzel sequences was known only to lie in an interval. The paper improves the Roselle-Stanton lower bound to match the pigeonhole upper bound in the…
For every n ≥ 2k + 1, is the independence polynomial of GP(n, k) real-rooted if and only if k is even? Exact Sturm counts refute both directions.
Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood conjectured that every graph of degree-d polynomial growth embeds into the strong product of d trees of linear growth and a bounded clique. False for d = 4: a…
White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank 9 binary matroid constitutes a counterexample.
Hamaker and Reiner conjectured that the order complex of an open interval (u,w) in the ASM weak order is contractible unless w is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere.…
A graph G is maximal non-Hamiltonian if it is non-Hamiltonian but G + e is Hamiltonian for every nonedge e. In 1994 Vu Dinh Hoa conjectured a property of G - V(C) for a longest cycle C of such a graph. Disproved by an explicit base graph…
Pak and Slonim conjectured that stretched Schubert structure constants are eventually polynomial. They are. Monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial, and the Schubert duality of…
A cyclic coloration of a triangulation of a closed 2-manifold gives the faces around every vertex distinct colors. Chen and Lawrencenko made two conjectures about the cyclic chromatic number of minimal triangulations in 1999. Their second…
Pach conjectured that n Jordan arcs, pairwise crossing exactly once with no triple points, have O(n) tangent pairs. The best known bound stood at O(n^7/4); the paper improves it to O(n^3/2) (and O(n^5/3) in the at-most-one-crossing…
For a graph G, its vertex deck is the multiset of graphs obtained by deleting one vertex. Bowler, Brown, and Fenner (BBF) proposed 2⌊(n−1)/3⌋ as the maximum possible overlap between the decks of two nonisomorphic n-vertex graphs, for all…
How dense can a sum-free subset of the lattice cube 1,…,n^d be? Aydinian and Cameron asked for the limiting density, which is also Problem 6 in Ben Green's list of 100 open problems. The natural conjecture is that the optimum is a slice x…
In the Frankl-Pach-Erdős circle of VC-dimension problems, the first arXiv version of the paper posed the k=3 case of a witness construction question. ChatGPT 5.4 Pro answered it; the published construction generalizes the model's response,…
Deng, Tidor and Zhao asked whether [N] admits a coloring with N^o(1) colors and no symmetrically coloured 4-term arithmetic progression, giving an O(N^log_223) coloring. The paper gives an O_k(N^4/k^2) coloring of [N] avoiding…
Reiner conjectured a description of the homotopy types of intervals in higher Bruhat orders. In corank 3 it holds: the facial intervals of B(n,n-3) are exactly the spherical intervals, and every other interval is contractible.
Simonovits conjectured that if a forbidden family F with p(F) > 1 has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of p graphs, each extremal for a family of chromatic number two.…
Let A(k) be the largest possible number of moves in a north-east lattice path whose visited vertices contain no k collinear points. Gerver (1979) and Gerver and Ramsey (1979) bounded A(k) by exp(Ω(log(k)^2)) ≤ A(k) ≤ exp(O(k^4)), and…
Korsky, Saffat and Aiylam bounded the growth constant c(G) for integer-valued Lipschitz functions on G(n,d/n) between 1/(2d) and 4log^2 d/d up to lower-order terms. The random-graph side is sharpened.