A new upper bound on the number of neighborly boxes in R^d

A new upper bound on the number of neighborly boxes in R^d is given. We apply a classical result of Kleitman on the maximum size of sets with a given diameter in discrete hypercubes. We also present results of some computational experiments and an emerging conjecture

Exact values and improved bounds on kk-neighborly families of boxes

A finite family $\mathcal{F}$ of $d$-dimensional convex polytopes is called $k$-neighborly if $d-k\le\textup{dim}(C\cap C')\le d-1$ for any two distinct members $C,C'\in\mathcal{F}$. In 1997, Alon initiated the study of the general function $n(k,d)$, which is defined to be the maximum size of $k$-neighborly families of standard boxes in $\mathbb{R}^{d}$. Based on a weighted count of vectors in $\{0,1\}^{d}$, we improve a recent upper bound on $n(k,d)$ by Alon, Grytczuk, Kisielewicz, and Przes\l awski for any positive integers $d$ and $k$ with $d\ge k+2$. In particular, when $d$ is sufficiently large and $k\ge 0.123d$, our upper bound on $n(k,d)$ improves the bound $\sum_{i=1}^{k}2^{i-1}\binom{d}{i}+1$ shown by Huang and Sudakov exponentially. Furthermore, we determine that $n(2,4)=9$, $n(3,5)=18$, $n(3,6)=27$, $n(4,6)=37$, $n(5,7)=74$, and $n(6,8)=150$. The stability result of Kleitman's isodiametric inequality plays an important role in the proofs.Comment: 17 pages. The main results were further improve

Clique versus Independent Set

Yannakakis' Clique versus Independent Set problem (CL-IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size $O(n^{\log n})$, and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a polynomial CS-separator almost surely exists for random graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a clique and a stable set) then there exists a constant $c_H$ for which we find a $O(n^{c_H})$ CS-separator on the class of H-free graphs. This generalizes a result of Yannakakis on comparability graphs. We also provide a $O(n^{c_k})$ CS-separator on the class of graphs without induced path of length k and its complement. Observe that on one side, $c_H$ is of order $O(|H| \log |H|)$ resulting from Vapnik-Chervonenkis dimension, and on the other side, $c_k$ is exponential. One of the main reason why Yannakakis' CL-IS problem is fascinating is that it admits equivalent formulations. Our main result in this respect is to show that a polynomial CS-separator is equivalent to the polynomial Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition into k complete bipartite graphs, then its chromatic number is polynomially bounded in terms of k. We also show that the classical approach to the stubborn problem (arising in CSP) which consists in covering the set of all solutions by $O(n^{\log n})$ instances of 2-SAT is again equivalent to the existence of a polynomial CS-separator

On the number of neighborly simplices in R^d

Two $d$-dimensional simplices in $R^d$ are neighborly if its intersection is a $(d-1)$-dimensional set. A family of $d$-dimensional simplices in $R^d$ is called neighborly if every two simplices of the family are neighborly. Let $S_d$ be the maximal cardinality of a neighborly family of $d$-dimensional simplices in $R^d$. Based on the structure of some codes $V\subset \{0,1,*\}^n$ it is shown that $\lim_{d\rightarrow \infty}(2^{d+1}-S_d)=\infty$. Moreover, a result on the structure of codes $V\subset \{0,1,*\}^n$ is given