Related papers: A note on the colorful fractional Helly theorem

Helly's selection theorem provides a criterion for compactness of sets of single-variable functions with bounded pointwise variation. Fra{\v{n}}kov{\'a} has given a proper extension of Helly's theorem to the setting of single-variable…

In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carath\'eodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type…

Consider a coloring of a graph such that each vertex is assigned a fraction of each color, with the total amount of colors at each vertex summing to $1$. We define the fractional defect of a vertex $v$ to be the sum of the overlaps with…

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…

A simplicial graph is said to be (coarsely) Helly if any collection of pairwise intersecting balls has non-empty (coarse) intersection. (Coarsely) Helly groups are groups acting geometrically on (coarsely) Helly graphs. Our main result is…

Chromatic polynomials have been studied extensively, giving us results such as the Fundamental Reduction Theorem and closed formulas for the chromatic polynomials of common classes of graphs. Though, none of those extend to the context of…

We prove the following sparse approximation result for polytopes. Assume that $Q$ is a polytope in John's position. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $Q \subseteq - 2d^2 \, Q'$. As a consequence,…

We generalize a result of Tibor Gallai as follows: for any finite set of points $\mathcal{S}$ in the plane, if the plane is colored in finitely many colors, then there exist $2^{\aleph_0}$ monochromatic subsets of the plane homothetic to…

This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical…

In this paper, we study the properties of the colored HOMFLY polynomials via HOMFLY skein theory. We prove some limit behaviors and symmetries of the colored HOMFLY polynomial predicted in some physicists' recent works.

We prove that for a topological space X with the property that $H_p(U)=0$ for $p\geq d$ and every open subset $U$ of $X$, a finite family of open sets in $X$ has nonempty intersection if for any subfamily of size $j$, $1\leq j \leq d+1$,…

In this paper, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out…

A well-known result of Burr, Erd\H{o}s and Spencer [Transactions of the American Mathematical Society, 1975] determines the $2$-colour Ramsey number for any sufficiently large collection of vertex-disjoint copies of a fixed graph $H$…

We show that for arbitrary linearly ordered set $X$ any bounded family of (not necessarily, continuous) real valued functions on $X$ with bounded total variation does not contain independent sequences. We obtain generalized Helly's…

In this paper, we extend the paraxial conical refraction model to the case of the partially coherent light using the unified optical coherence theory. We demonstrate the decomposition of conical refraction correlation functions into…

One widely applied sufficient condition for the existence of a colorful simplex in a vertex-colored simplicial complex is a topological extension of Hall's transversal theorem due to Aharoni, Haxell, and Meshulam. We prove a similar…

We prove the analogue of Helly's theorem for systolic complexes. Namely, we show that 7-systolic complexes have Helly dimension less or equal to 1, whereas 6-systolic complexes have Helly dimension bounded from the above by 2.

In fuzzy group theory many versions of the well-known Lagrange's theorem have been studied. The aim of this article is to investigate the converse of one of those results. This leads to an interesting characterization of finite cyclic…

The Hall ratio of a graph $G$ is the maximum value of $v(H) / \alpha(H)$ taken over all non-null subgraphs $H$ of $G$. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number. In this note, we present various…

A d-partite hypergraph is called *fractionally balanced* if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall's theorem we prove…