Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.

Abstract not found

Abstract. A word w is called synchronizing (recurrent, reset, directed) word of a deterministic finite automaton (DFA) if w sends all states of the automaton on a unique state. Jan Černy had found in 1964 a sequence of n-state complete DFA with shortest synchronizing word of length (n − 1) 2. He had conjectured that it is an upper bound for the length of the shortest synchronizing word for any n-state complete DFA. The examples of DFA with shortest synchronizing word of length (n−1) 2 are relatively rare. To the Černy sequence were added in all examples of

We prove an upper bound, tight up to a factor of 2, for the number of vertices of level at most ℓ in an arrangement of n halfspaces in R d, for arbitrary n and d (in particular, the dimension d is not considered constant). This partially settles a conjecture of Eckhoff, Linhart, and Welzl. Up to the factor of 2, the result generalizes McMullen’s Upper Bound Theorem for convex polytopes (the case ℓ = 0) and extends a theorem of Linhart for the case d ≤ 4. Moreover, the bound sharpens asymptotic estimates obtained by Clarkson and Shor. The proof is based on the h-matrix of the arrangement (a generalization, introduced by Mulmuley, of the h-vector of a convex polytope). We show that bounding appropriate sums of entries of this matrix reduces to a lemma about quadrupels of sets with certain intersection properties, and we prove this lemma, up to a factor of 2, using tools from multilinear algebra. This extends an approach of Alon and Kalai, who used linear algebra methods for an alternative proof of the classical Upper Bound Theorem. The bounds for the entries of the h-matrix also imply bounds for the number of i-dimensional faces, i> 0, at level at most ℓ. Furthermore, we discuss a connection with crossing numbers of graphs that was one of the main motivations for investigating exact bounds that are valid for arbitrary dimensions. 1.

Given an r-graph G on [n], we are allowed to add consecutively new edges to it provided that every time a new r-graph with at least l edges and at most m vertices appears. Suppose we have been able to add all edges. What is the minimum number of edges in the original graph? For all values of parameters, we present an example of G which we conjecture to be minimum and establish the validity of our conjecture for a range of parameters. Our proof utilises count matroids which is a new family of matroids naturally extending that of White and Whiteley [15]. We characterise, in certain cases, the extremal graphs. In particular, we answer a question by Erd}os, Furedi and Tuza [3]. 1 Uniform Families A k-graph H is, as usual, a pair (V (H); E(H)) (vertices and edges) where E(H) is a subset of V (H) (k) = fA V (H) : jAj = kg. The size of H is e(H) = jE(H)j and the order is v(H) = jV (H)j. When isolated vertices don't matter, we usually identify H with E(H); then, for example, jH...

Abstract. A word w is called synchronizing (recurrent, reset, directable) word of deterministic finite automaton (DFA) if w brings all states of the automaton to an unique state. Černy conjectured in 1964 that every nstate synchronizable automaton possesses a synchronizing word of length at most (n − 1) 2. The problem is still open. It will be proved that the minimal length of synchronizing word is not greater than (n − 1) 2 /2 for every n-state (n> 2) synchronizable DFA with transition monoid having only trivial subgroups (such automata are called aperiodic). This important class of DFA accepting precisely star-free languages was involved and studied by Schŭtzenberger. So for aperiodic automata as well as for automata accepting only star-free languages, the Čern´y conjecture holds true. Some properties of an arbitrary synchronizable DFA and its transition semigroup were established.

A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Čern´y conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n−1) 2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n − 1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Čern´y conjecture holds true.

A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented. Some consequences on the length of the synchronizing word are discussed.

... Our main technique G-proof, developed from ideas of Kalai, is based on matroids derived from exterior algebra. We prove that if some graphs admit G-proof then so do `cones' and `joins' of such graphs, so that the w-satfunction can be computed for many graphs that are built up from certain elementary graphs by these operations.

doi 10.1287/moor.1050.0161