Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.36566

Helly theorems for lattice of flats of a matroid

Amir Jafari·Kiumars Kaveh
View original source ↗Read source document ↗Export metadata ↓

Abstract

A basis of a matroid is adapted to a flat if it contains a basis of the flat. We prove that a family of flats in a rank r matroid has a common adapted basis whenever every subfamily of at most r+1 flats does. The ground set and the family may be infinite; only the rank must be finite. This result, in particular, contains Klyachko's vector space Helly theorem for subspaces. As an application, we obtain a splitting criterion for tropical toric vector bundles on projective spaces. As a secondary application, we derive quadratic bounds for the Helly number for distributive generation in the lattice of flats.

Explore related subjects

math.COmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Amir Jafari, Kiumars Kaveh. 2026-10-27. Helly theorems for lattice of flats of a matroid. https://arxiv.org/abs/2610.36566

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A spanning tree model for chromatic homology

After the discovery of Khovanov homology, which categorifies the Jones polynomial, an analogous categorification of the chromatic polynomial, known as chromatic homology, was introduced. Its graded Euler characteristic recovers the chromatic polynomial. In this paper, we present a spanning tree model for the chromatic complex, i.e., we describe a chain complex generated by certain spanning trees of the graph that is chain homotopy equivalent to the chromatic complex. We employ the spanning tree model over $\mathcal{A}_m:= \frac{\mathbb{Z}[x]}{ }$ algebra to answer two open questions. First, we establish the conjecture posed by Sazdanovic and Scofield regarding the homological span of chromatic homology over $\mathcal{A}_m$ algebra, demonstrating that for any graph $G$ with $v$ vertices and $b$ blocks, the homological span is $v - b$. Additionally, we prove a conjecture of Helme-Guizon, Przytycki, and Rong concerning the existence of torsion of order dividing $m$ in chromatic homology over $\mathcal{A}_m$ algebra.

math.CO↗
arXiv2026

Total eight-colorings of maximal planar graphs of maximum degree six

We prove that every finite simple maximal planar graph of maximum degree at most six admits a total eight-coloring. We first show that every proper vertex four-coloring of a subcubic graph extends to a total five-coloring. A curvature inequality for boundary walks then gives a matching decomposition of four-connected triangulations with degrees five and six, extending their prescribed vertex four-colorings to total eight-colorings. In a minimum counterexample, two local boundary colorings force the degree-four vertices to have disjoint neighborhoods. A characterization of the local extension obstructions permits these vertices to be restored simultaneously. The proof uses Hamiltonian-connectedness of four-connected planar graphs and Kardoš's Hamiltonicity theorem for three-connected cubic planar graphs.

math.CO↗
arXiv2026

Two families of Young tableaux with periodic walls: Counting formulas and differential transcendence

Banderier, Marchal, and Wallner studied the enumeration problem for Young tableaux with walls. These are similar to standard Young tableaux, except that local decreases are allowed at some walls. In 2021, Banderier and Wallner counted Young tableaux with periodic walls, in which the elementary blocks are Young diagrams of shape $(2,2)$. They obtained exact counting formulas for $30$ of the $32$ classes up to symmetry. In this paper, we first derive recurrences for counting the remaining two classes of Young tableaux with periodic walls. We then prove that their counting exponential generating functions are differentially transcendental. Consequently, they are neither D-algebraic nor D-finite. This disproves a conjecture of Banderier and Wallner.

math.CO↗