294 papers
This paper establishes that finite chains within the two-branching countable ultrahomogeneous pseudotree possess finite big Ramsey degrees, specifically determining the degree for chains of length two to be seven, thereby presenting the first example of a countable ultrahomogeneous structure in a finite language where some finite substructures have finite big Ramsey degrees while others have infinite ones.
This paper establishes that for hypergraphs derived from cross-free connected subgraphs of a bounded-genus host graph, there exists a support graph that also maintains bounded genus, thereby generalizing prior planar results and providing a unified framework for solving packing, covering, and coloring problems on surfaces.
This paper introduces a new scoring variant of the combinatorial game Nim that generalizes both normal and misère play rules, and analyzes its theoretical properties, including optimal strategies and payoff functions.
This paper introduces the concept of Lorentzian proper position to characterize elementary quotients of M-convex functions and analyze the incidence geometry of tropical linear spaces, revealing that while certain classical incidence properties fail for general tropical spaces, they hold for those possessing adjoints.
This paper investigates permutation classes formed by finite subpermutations of infinite pin sequences, proving that they possess proper growth rates and providing a method to calculate them.
This paper establishes an explicit formula for the rank of the -walk matrix of the Dynkin graph and proves that its Smith normal form consists of a single 1, followed by twos, and the remaining entries being zero.
This paper introduces the dib-chromatic number as an extension of the -chromatic number to directed graphs via acyclic vertex colorings, providing general bounds and specific results for tournaments and regular digraphs.
This paper establishes a tropical analogue of Porteous' formula by developing characteristic classes for tropical vector bundles on rational polyhedral spaces with boundary, utilizing a splitting principle and the boundary framework to ensure degeneracy loci achieve their expected codimension.
This paper explores invariants of almost embeddings of graphs in the plane by establishing relations among them, connecting these to the homology of the deleted product, constructing examples, and presenting these topological concepts in an accessible manner while highlighting open conjectures.
This paper investigates the minimum number of comparability subgraphs required to partition the edges of perfect graphs, demonstrating that while most perfect graph classes (and almost all perfect graphs) can be partitioned into at most two such subgraphs, interval graphs may require an arbitrarily large number.
This paper characterizes graphs whose edge ideals possess Scarf resolutions as exactly the gap-free forests, while also classifying connected graphs where all powers of their edge ideals admit Scarf resolutions and providing recursive constructions for these complexes.
This paper establishes that the rook polynomial of grid polyominoes coincides with the h-polynomial of their corresponding coordinate rings by utilizing simplicial complex theory to generalize previous findings on frame polyominoes.
This paper improves upon previous results by Heise et al. by proving that the weak chromatic number of -uniform hypergraphs arising from linearly or PL-embeddable simplicial complexes in is infinite for most dimensions and uniformities, and further extends these findings to show that the weak chromatic number of -dimensional faces in triangulations of any fixed triangulable -manifold is infinite for all $1 \le s \le d$.
This paper introduces a simplified perspective on Hodge-Newton indecomposability to provide a uniform proof of a combinatorial identity derived from affine Deligne-Lusztig varieties with finite Coxeter parts.
This paper establishes that the decomposition theorem components for contraction maps of torus actions of complexity one are intersection cohomology complexes of even codimensional subvarieties, leading to the vanishing of odd-dimensional intersection cohomology for rational complete varieties of this type and providing explicit formulas for the Betti numbers of affine trinomial hypersurfaces based on their defining equations.
This paper investigates the distribution of boundary points of expansion to establish a lower bound on the mutual distances between runners on a circular track under specific equidistant spacing conditions, thereby providing a partial application toward the Lonely Runner Conjecture.
This paper investigates the WELLDOC property, an abelian-type characteristic of infinite words concerning the distribution of Parikh vectors of prefixes, and establishes a criterion for determining whether words generated by morphisms possess this property.
This paper presents a neurosymbolic collaboration between an LLM-powered agent, symbolic computation tools, and human researchers that successfully discovered and formally verified a new tight lower bound on the imbalance of Latin squares for the case , demonstrating the potential of AI-human partnerships in pure mathematical discovery.
This paper presents a concise and self-contained proof of Delahan's theorem, demonstrating that every simple graph with vertices appears as an induced subgraph of a Steinhaus graph with vertices by utilizing the concept of generating index sets for Steinhaus triangles.
This paper proves that every large connected Cayley graph with degree at least for some absolute constant contains a Hamilton cycle, thereby advancing the Lovász conjecture by improving previous density thresholds through a proof that utilizes an arithmetic regularity lemma tailored to Cayley graphs instead of Szemerédi's regularity lemma.