338 papers
This paper investigates the "evil twin property" in sums of wildflowers and mutant flowers within combinatorial game theory, identifying a maximal closed set of such games that possess this property and proving that determining their outcome classes is NP-hard.
This paper establishes a deep connection between convex neural codes and oriented matroids through categorical functors and geometric partial orders, ultimately proving that determining whether a combinatorial code is convex is an NP-hard problem by leveraging Mnëv-Sturmfels universality.
This paper investigates triangular line arrangements on the projective plane by proving their combinatorics are always realized by Roots-of-Unity-Arrangements, establishing conditions for their freeness, and demonstrating that freeness is not determined solely by weak combinatorics.
This paper introduces a comprehensive generalization of the switch operation on -graphs, axiomatically studying its impact on homomorphisms to resolve open problems regarding categorical products and chromatic numbers, including explicit constructions and calculations for forests.
Using the compression method, this paper generalizes lower bounds for the Erdős unit distance and distinct distance problems to higher dimensions ( for ), providing alternative proofs and establishing new estimates for the number of distinct distances.
This paper proves the nonexistence of Heronian triangles with three integer medians by deriving a universal identity for arbitrary triangles and establishing a lemma demonstrating that such triangles, if they existed, would necessarily occur in non-similar pairs.
This paper analyzes a two-player permutation game inspired by the Erdős-Szekeres Theorem, determining the winner and providing a winning strategy for cases where the parameter is greater than or equal to and is an integer between 2 and 5.
This paper presents the construction of a 2-distance set consisting of 277 points in 23-dimensional Euclidean space, where the pairwise distances between points are restricted to the values 2 and .
The paper establishes that any collection of hyperplanes covering the vertices of an -dimensional hypercube, where each vertex is covered by a hyperplane involving every coordinate variable, must contain at least hyperplanes, a result that generalizes the skew covers problem and yields tight bounds for slicing hypercube edges with bounded integer coefficients.
This paper establishes elementary upper and lower bounds for the partition function and its generalizations by applying a geometric inequality in Euclidean space.
This paper proposes and rigorously establishes a symmetrization relation between 4d BPS quivers and 3d symmetric quivers, demonstrating that the wall-crossing structure of 4d Argyres-Douglas theories is isomorphic to the unlinking of their 3d counterparts and that these symmetric quivers successfully capture the Schur indices of the original 4d theories.
This paper establishes the NP-completeness of determining whether graphs lacking efficient edge dominating sets possess at least two perfect edge dominating sets, while also presenting a cubic-time algorithm to find, count, and weight minimal perfect edge dominating sets in -free graphs.
This paper proves Clark Kimberling's conjectures regarding his 2017 binary sequence by utilizing the Walnut theorem-prover to establish its connection to the infinite Tribonacci word and to determine its subword complexity and critical exponent.
This paper provides a complete characterization of all model category structures on finite lattices by utilizing transfer systems, thereby establishing new connections between abstract homotopy theory and equivariant methods.
This paper confirms Xin and Zhang's conjecture by deriving a simple product formula for the characteristic polynomial of a specific tridiagonal matrix related to the enumeration of nonnegative integer matrices with fixed row and column sums, while also extending these results to broader families of such matrices.
This paper extends previous results on the sum-product problem by proving that any set of 10 or 11 natural numbers must generate at least 30 or 34 distinct pairwise sums or products, respectively, while providing a classification of extremal sets and analyzing additive structures in small subsets of generalized geometric progressions.
This paper investigates the relationship between perfect divisibility and perfect weighted divisibility in graphs, demonstrating that $2P_3$-free or claw-free minimally nonperfectly divisible graphs contain no clique cutset, thereby providing a conditional answer to a question posed by Hoàng.
This paper proves the bijection between the faces of the cosmohedron and Matryoshkas, elucidates the combinatorial structure of these generalized chiseled polytopes, and outlines their application to ultraviolet divergences in loop-integrated Feynman amplitudes.
This paper establishes that for any integer and any greater than the unique positive root of a specific polynomial, the restricted -fold sumset of a subset in a finite abelian group of sufficiently large odd order equals the entire group whenever , thereby generalizing previous results on cyclic groups and identifying as the optimal asymptotic density threshold.
This paper provides combinatorial characterizations of virtually torsion-free and virtually free groups by utilizing canonical graph decomposition theory to establish necessary and sufficient conditions involving the structural properties of -local covers and their associated tree-decompositions.