Color critical hypergraphs and forbidden configurations

Abstract

International audience The present paper connects sharpenings of Sauer's bound on forbidden configurations with color critical hypergraphs. We define a matrix to be \emphsimple if it is a $(0,1)-matrix$ with no repeated columns. Let $F$ be $a k× l (0,1)-matrix$ (the forbidden configuration). Assume $A$ is an $m× n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define $forb(m,F)$ as the best possible upper bound on n, for such a matrix $A$, which depends on m and $F$. It is known that $forb(m,F)=O(m^k)$ for any $F$, and Sauer's bond states that $forb(m,F)=O(m^k-1)$ fore simple $F$. We give sufficient condition for non-simple $F$ to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.

Embedding graphs into larger graphs: results, methods, and problems

“…We skip here the topic of colour-critical hypergraphs, listing just a few fascinating results on them: e.g., Abbott and Liu [2] and results of Krivelevich [556]. Deuber, Kostochka, Sachs [238], Rödl and Siggers [703] and Anstee, Fleming, Füredi and Sali [50,384]. Toft [806], and Simonovits [752], Rödl and Tuza [706], Sachs and Stiebitz [711,712,713] Stiebitz, [782] Kostochka and Stiebitz [552], Stiebitz, Tuza, and Voigt [783].…”

Information Width

“…The previous representation based on (1) is subsumed by this representation since instead of X we have Z and the sets Y x , x ∈ Π X (A) defined in (2) are now represented by the sets Y z , z ∈ Z x . Since X indexes all possible properties of sets in Y then for any A as in (1) there exists an x ∈ X in the new representation such that Π X (A) in Kolmogorov's representation is equivalent to Z x in the new representation.…”

“…It has also been shown to relate to various fundamental results in different fields, e.g., statistical learning theory [26], combinatorial geometry [20], graph theory [3,10] and the theory of empirical processes [22]. It is a member of a more general class of properties that are expressed in terms of certain allowed or forbidden restrictions [1]. In this paper we focus on properties based on the trace of a class which are expressed in terms of a positive integer parameter d in the following general form:…”

“…It contains k elements if and only if cols(U ) consists of distinct elements, or less than k elements if two columns are the same. Denote by S a simple binary matrix as one all of whose columns are distinct ( [1]). We claim that the conditional distribution of the set of columns of a random binary matrix, knowing that the matrix is simple, is the uniform probability distribution P * n,k .…”