A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are n^{1/2} lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are (n/3)^{1/2} of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs

Semi-algebraic Ramsey numbers

Given a finite point set $P \subset \mathbb{R}^d$, a $k$-ary semi-algebraic relation $E$ on $P$ is the set of $k$-tuples of points in $P$, which is determined by a finite number of polynomial equations and inequalities in $kd$ real variables. The description complexity of such a relation is at most $t$ if the number of polynomials and their degrees are all bounded by $t$. The Ramsey number $R^{d,t}_k(s,n)$ is the minimum $N$ such that any $N$-element point set $P$ in $\mathbb{R}^d$ equipped with a $k$-ary semi-algebraic relation $E$, such that $E$ has complexity at most $t$, contains $s$ members such that every $k$-tuple induced by them is in $E$, or $n$ members such that every $k$-tuple induced by them is not in $E$. We give a new upper bound for $R^{d,t}_k(s,n)$ for $k\geq 3$ and $s$ fixed. In particular, we show that for fixed integers $d,t,s$, $R^{d,t}_3(s,n) \leq 2^{n^{o(1)}},$ establishing a subexponential upper bound on $R^{d,t}_3(s,n)$. This improves the previous bound of $2^{n^C}$ due to Conlon, Fox, Pach, Sudakov, and Suk, where $C$ is a very large constant depending on $d,t,$ and $s$. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in $\mathbb{R}^d$. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results

Geometry of contextuality from Grothendieck's coset space

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli operators [Information 5, 209 (2014); 1310.4267 and 1404.6986 (quant-ph)]. Now we find that the non-existence of a dessin for which the commutator (a, b) = a^ (--1) b^( --1) ab precisely corresponds to the commutator of quantum observables [A, B] = AB -- BA on all lines of the geometry is a signature of quantum contextuality. This occurs first at index |G : H| = 9 in Mermin's square and at index 10 in Mermin's pentagram, as expected. Commuting sets of n-qubit observables with n \textgreater{} 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.Comment: 13 pages, Quant. Inf. Pro

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Discrete Geometry

[no abstract available

Higher-order Erdos--Szekeres theorems

Let P=(p_1,p_2,...,p_N) be a sequence of points in the plane, where p_i=(x_i,y_i) and x_1<x_2<...<x_N. A famous 1935 Erdos--Szekeres theorem asserts that every such P contains a monotone subsequence S of $\sqrt N$ points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of $\Omega(\log N)$ points. Monotonicity is a property determined by pairs of points, and convexity concerns triples of points. We propose a generalization making both of these theorems members of an infinite family of Ramsey-type results. First we define a (k+1)-tuple $K\subseteq P$ to be positive if it lies on the graph of a function whose kth derivative is everywhere nonnegative, and similarly for a negative (k+1)-tuple. Then we say that $S\subseteq P$ is kth-order monotone if its (k+1)-tuples are all positive or all negative. We investigate quantitative bound for the corresponding Ramsey-type result (i.e., how large kth-order monotone subsequence can be guaranteed in every N-point P). We obtain an $\Omega(\log^{(k-1)}N)$ lower bound ((k-1)-times iterated logarithm). This is based on a quantitative Ramsey-type theorem for what we call transitive colorings of the complete (k+1)-uniform hypergraph; it also provides a unified view of the two classical Erdos--Szekeres results mentioned above. For k=3, we construct a geometric example providing an $O(\log\log N)$ upper bound, tight up to a multiplicative constant. As a consequence, we obtain similar upper bounds for a Ramsey-type theorem for order-type homogeneous subsets in R^3, as well as for a Ramsey-type theorem for hyperplanes in R^4 recently used by Dujmovic and Langerman.Comment: Contains a counter example of Gunter Rote which gives a reply for the problem number 5 in the previous versions of this pape