A Tutte polynomial inequality for lattice path matroids

Let $M$ be a matroid without loops or coloops and let $T(M;x,y)$ be its Tutte polynomial. In 1999 Merino and Welsh conjectured that $$\max(T(M;2,0), T(M;0,2))\geq T(M;1,1)$$ holds for graphic matroids. Ten years later, Conde and Merino proposed a multiplicative version of the conjecture which implies the original one. In this paper we prove the multiplicative conjecture for the family of lattice path matroids (generalizing earlier results on uniform and Catalan matroids). In order to do this, we introduce and study particular lattice path matroids, called snakes, used as building bricks to indeed establish a strengthening of the multiplicative conjecture as well as a complete characterization of the cases in which equality holds.Comment: 17 pages, 9 figures, improved exposition/minor correction

Triangle areas in line arrangements

A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between $\Omega(n^2)$ and $O(n^{9/4})$. This result is strongly connected to both additive combinatorial results and Szemer\'edi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.Comment: Title is shortened. Some typos and small errors were correcte

On a colorful problem by Dol'nikov concerning translates of convex bodies

In this note we study a conjecture by Jer\'onimo-Castro, Magazinov and Sober\'on which generalized a question posed by Dol'nikov. Let $F_1,F_2,\dots,F_n$ be families of translates of a convex compact set $K$ in the plane so that each two sets from distinct families intersect. We show that, for some $j$, $\bigcup_{i\neq j}F_i$ can be pierced by at most $4$ points. To do so, we use previous ideas from Gomez-Navarro and Rold\'an-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square

Codimension two and three Kneser Transversals

Let $k,d,\lambda \geqslant 1$ be integers with $d\geqslant \lambda$ and let $X$ be a finite set of points in $\mathbb{R}^{d}$. A $(d-\lambda)$-plane $L$ transversal to the convex hulls of all $k$-sets of $X$ is called Kneser transversal. If in addition $L$ contains $(d-\lambda)+1$ points of $X$, then $L$ is called complete Kneser transversal.In this paper, we present various results on the existence of (complete) Kneser transversals for $\lambda =2,3$. In order to do this, we introduce the notions of stability and instability for (complete) Kneser transversals. We first give a stability result for collections of $d+2(k-\lambda)$ points in $\mathbb{R}^d$ with $k-\lambda\geqslant 2$ and $\lambda =2,3$. We then present a description of Kneser transversals $L$ of collections of $d+2(k-\lambda)$ points in $\mathbb{R}^d$ with $k-\lambda\geqslant 2$ for $\lambda =2,3$. We show that either $L$ is a complete Kneser transversal or it contains $d-2(\lambda-1)$ points and the remaining $2(k-1)$ points of $X$ are matched in $k-1$ pairs in such a way that $L$ intersects the corresponding closed segments determined by them. The latter leads to new upper and lower bounds (in the case when $\lambda =2$ and $3$) for $m(k,d,\lambda)$ defined as the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ admit a Kneser transversal.Finally, by using oriented matroid machinery, we present some computational results (closely related to the stability and unstability notions). We determine the existence of (complete) Kneser transversals for each of the $246$ different order types of configurations of $7$ points in $\mathbb{R}^3$

Triangle areas in line arrangements

A widely investigated subject in combinatorial geometry, originated from Erd}os, is the following. Given a point set P of cardinality n in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. What is the maximum number of triangles of unit area, maximum area or minimum area, that can be determined by an arrangement of n lines in the plane? We prove that the order of magnitude for the maximum occurrence of unit areas lies between Omega(n^2) and O(n^9/4+epsilon), for every epsilon > 0. This result is strongly connected to additive combinatorial results and Szemeredi-Trotter type incidence theorems. Next we show an almost tight bound for the maximum number of minimum area triangles. Finally, we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, $$\frac{40\lfloor (k-1)/2 \rfloor -8}{19} \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11.$$ Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.Comment: 23 pages, 11 figures, to appear at Discrete Mathematic