Starting with a result in combinatorial number theory we prove that (apart from a couple of exceptions that can be classified), for any elements a 1 , . . . ,a n of GF(q), there are distinct field elements b 1 , . . . , b n such that a 1 b 1 +· · ·+a n b n = 0. This implies the classification of hyperplanes lying in the union of the hyperplanes X i = X j in a vector space over GF(q), and also the classification of those multisets for which all reduced polynomials of this range are of reduced degree q − 2. The proof is based on the polynomial method.

LetHbe a graph onnvertices and let the blow-up graphG[H] be defined as follows. We replace each vertexviofHby a clusterAiand connect some pairs of vertices ofAiandAjif (vi,vj) is an edge of the graphH. As usual, we define the edge density betweenAiandAjasWe study the following problem. Given densities γijfor each edge (i,j) ∈E(H), one has to decide whether there exists a blow-up graphG[H], with edge densities at least γij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic toH,i.e., noHappears as a transversal inG[H]. We calldcrit(H) the maximal value for which there exists a blow-up graphG[H] with edge densitiesd(Ai,Aj)=dcrit(H) ((vi,vj) ∈E(H)) not containingHin the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.First, in the case of treeTwe give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversalTin the blow-up graph. Then we give general bounds ondcrit(H) in terms of the maximal degree. In connection with the extremal structure, the so-called star decomposition is proved to give the best construction forH-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools.

In this paper we propose a multipartite version of the classical Turán problem of determining the minimum number of edges needed for an arbitrary graph to contain a given subgraph. As it turns out, here the non-trivial problem is the determination of the minimal edge density between two classes that implies the existence of a given subgraph. We determine the critical edge density for trees and cycles as forbidden subgraphs, and give the extremal structure. Surprisingly, this critical edge density is strongly connected to the maximal eigenvalue of the graph. Furthermore, we give a sharp upper and lower bound in general, in terms of the maximum degree of the forbidden graph.

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Abstract:We study the existence and the number of k-dominating independent sets in certain graph families. While the case k = 1 namely the case of maximal independent sets-which is originated from Erdős and Moser-is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k-dominating independent sets in n-vertex graphs is between c k · 2k

A conjecture of Erdős from 1967 asserts that any graph on n vertices which does not contain a fixed r-degenerate bipartite graph F has at most Cn 2−1/r edges, where C is a constant depending only on F . We show that this bound holds for a large family of r-degenerate bipartite graphs, including all r-degenerate blow-ups of trees. Our results generalise many previously proven cases of the Erdős conjecture, including the related results of Füredi and Alon, Krivelevich and Sudakov. Our proof uses supersaturation and a random walk on an auxiliary graph.

Let Π q be an arbitrary finite projective plane of order q. A subset S of its points is called saturating if any point outside S is collinear with a pair of points from S. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to ⌈ √ 3q ln q⌉ + ⌈( √ q + 1)/2⌉. The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. Built on a recent idea of Karasev and Petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner. The main consequence is the proof of a conjecture of Forrester related to the Calogero--Sutherland model. In fact we prove a more general theorem, which includes Aomoto's constant term identity at the same time. We also demonstrate the relevance of the method in additive combinatorics.Comment: 21 page