A coding problem for pairs of subsets

Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap B_2=\emptyset$). Define the distance of these pairs by $d(\{A_1,A_2\} ,\{B_1,B_2\})=\min \{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\}$. This is the minimum number of elements of $A_1\cup A_2$ one has to move to obtain the other pair $\{B_1,B_2\}$. Let $C(n,k,d)$ be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs is at least $d$. Here we establish a conjecture of Brightwell and Katona concerning an asymptotic formula for $C(n,k,d)$ for $k,d$ are fixed and $n\to \infty$. Also, we find the exact value of $C(n,k,d)$ in an infinite number of cases, by using special difference sets of integers. Finally, the questions discussed above are put into a more general context and a number of coding theory type problems are proposed.Comment: 11 pages (minor changes, and new citations added

New Type of Coding Problem Motivated by Database Theory

The present paper is intended to survey the interaction between relational database theory and coding theory. In particular it is shown how an extremal problem for relational databases gives rise to a new type of coding problem. The former concerns minimal representation of branching dependencies that can be considered as a data mining type question. The extremal configurations involve d-distance sets in the space of disjoint pairs of k-element subsets of an n-element set X. Let X be an n-element finite set, 0 < k < n/2 an integer. Suppose that {A(1), B-1} and {A(2), B-2} are pairs of disjoint k-element subsets of X (that is, \A(1)\ = \B-1\ = \A(2)\ = \B-2\ = k, A(1) boolean AND B-1 = 0, A(2) boolean AND B-2 = 0). Define the distance of these pairs by d({A(1), B-1}, {A(2), B-2}) = min{\A(1) - A(2)\ + \B-1 - B-2\, \A(1) - B-2\ + \B-1 - A(2)\). (C) 2004 Elsevier B.V. All rights reserved

On some problems in extremal, probabilistic and enumerative combinatorics

This is a study of a small selection of problems from various areas of Combinatorics and Graph Theory, a fast developing field that provides a diverse spectrum of powerful tools with numerous applications to computer science, optimization theory and economics. In this thesis, we focus on extremal, probabilistic and enumerative problems in this field. A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family \F\subseteq \P(n) that does not contain a $2$-chain $F_1\subsetneq F_2$. Erd\H{o}s later extended this result and determined the largest family not containing a $k$-chain $F_1\subsetneq \ldots \subsetneq F_k$. Erd\H{o}s and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result. In Chapter 2 we answer their question for all families of size at most (1-\eps)2^n, provided $n$ is sufficiently larger compared to $k$ and \eps. The result of Chapter 2 is an example of a supersaturation, or Erd\H{o}s--Rademacher type result, which seeks to answer how many forbidden objects must appear in a set whose size is larger than the corresponding result. These supersaturation results are a key ingredient to a very recently discovered proof method, called the Container method. Chapters 3 and 4 show various examples of this method in action. In Chapter 3 we, among others, give tight bounds on the logarithm of the number of $t$-error correcting codes and illustrate how the Container method can be used to prove random analoges of classical extremal results. In Chapter 4 we solve a conjecture of Burosch--Demetrovics--Katona--Kleitman--Sapozhenko about estimating the number of families in $\{0,1\}^n$ which do not contain two sets and their union. In Chapter 5 we improve an old result of Erd\H{o}s and Spencer. Folkman's theorem asserts that for each $k \in \N$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-colored, there exists a set $A \subset [n]$ of size $k$ with the property that all the sums of the form $\sum_{x \in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the same color. In 1989, Erd\H{o}s and Spencer showed that $F(k) \ge 2^{ck^2/ \log k}$, where $c >0$ is an absolute constant; here, we improve this bound significantly by showing that $F(k) \ge 2^{2^{k-1}/k}$ for all $k\in \N$. Fox--Grinshpun--Pach showed that every $3$-coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $\Omega\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. In Chapter 6 we show, amongst others, that every such coloring contains a $2$-colored subgraph with chromatic number at least $n^{2/3}$, and this is best possible. As a direct corollary of our result we obtain a generalisation of the celebrated theorem of Erd\H{o}s-Szekeres, which states that any sequence of $n$ numbers contains a monotone subsequence of length at least $\sqrt{n}$