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

A constructive proof of the general Lovasz Local Lemma

The Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Simplifications of his procedure and relaxations of its restrictions were subsequently exhibited in several publications [Alo91, MR98, CS00, Mos06, Sri08, Mos08]. In [Mos09], a constructive proof was presented that works under negligible restrictions, formulated in terms of the Bounded Occurrence Satisfiability problem. In the present paper, we reformulate and improve upon these findings so as to directly apply to almost all known applications of the general Local Lemma.Comment: 8 page

Conflict-free coloring of graphs

We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.Comment: 12 page

The visible perimeter of an arrangement of disks

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n^1/2), then there is a stacking order for which the visible perimeter is Omega(n^2/3). We also show that this bound cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n^3/4) with respect to any stacking order. This latter bound cannot be improved either. Finally, we address the case where no more than c disks can have a point in common. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.Comment: 12 pages, 5 figure