Cross-intersecting families of vectors

Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we select $x \in A$ and $y \in B$, there are at least $r$ distinct indices $i$ such that $x_i = y_i$. We determine the maximum value of $|A|\cdot|B|$ over all pairs of $r$- cross-intersecting families and characterize the extremal pairs for $r \ge 1$, provided that $\min p_i >r+1$. The case $\min p_i \le r+1$ is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and answers a question of Zhang

Decomposition of multiple packings with subquadratic union complexity

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function $f(n)=o(n^2)$ with the property that any $n$ members of $\mathcal{X}$ determine at most $f(n)$ holes, which means that the complement of their union has at most $f(n)$ bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that $\mathcal{X}$ can be decomposed into at most $p$ ($1$-fold) packings, where $p$ is a constant depending only on $k$ and $f$.Comment: Small generalization of the main result, improvements in the proofs, minor correction

Every graph admits an unambiguous bold drawing

Let r and w be fixed positive numbers, w < r. In a bold drawing of a graph, every vertex is represented by a disk of radius r, and every edge by a narrow rectangle of width w. We solve a problem of van Kreveld [10] by showing that every graph admits a bold drawing in which the region occupied by the union of the disks and rectangles representing the vertices and edges does not contain any disk of radius r other than the ones representing the vertices. © 2015, Brown University. All rights reserved

On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves

A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any $n$ simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least $(1-o(1))n^2$. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let $S$ be a family of the graphs of $n$ continuous real functions defined on $\mathbb{R}$, no three of which pass through the same point. If there are $nt$ pairs of touching curves in $S$, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$.Comment: To appear in SODA 201