Hipergráfok = Hypergraphs

A projekt célkitűzéseit sikerült megvalósítani. A négy év során több mint száz kiváló eredmény született, amiből eddig 84 dolgozat jelent meg a téma legkiválóbb folyóirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos régóta fennálló sejtést bebizonyítottunk, egész régi nyitott problémát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolódó területeken. A problémák némelyike sok éve, olykor több évtizede nyitott volt. Nem egy közvetlen kutatási eredmény, de szintén bizonyos értékmérő, hogy a résztvevők egyike a Norvég Királyi Akadémia tagja lett és elnyerte a Steele díjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize

Independent transversals in locally sparse graphs

Let G be a graph with maximum degree \Delta whose vertex set is partitioned into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G) containing exactly one vertex from each part V_i. If it is also an independent set, then we call it an independent transversal. The local degree of G is the maximum number of neighbors of a vertex v in a part V_i, taken over all choices of V_i and v \not \in V_i. We prove that for every fixed \epsilon > 0, if all part sizes |V_i| >= (1+\epsilon)\Delta and the local degree of G is o(\Delta), then G has an independent transversal for sufficiently large \Delta. This extends several previous results and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalize this result to transversals that induce no cliques of size s. (Note that independent transversals correspond to s=2.) In that context, we prove that parts of size |V_i| >= (1+\epsilon)[\Delta/(s-1)] and local degree o(\Delta) guarantee the existence of such a transversal, and we provide a construction that shows this is asymptotically tight.Comment: 16 page

Point selections and weak e-nets for convex hulls

One of our results: let X be a finite set on the plane, 0 < g < 1, then there exists a set F (a weak g-net) of size at most 7/e 2 such that every convex set containing at least e\X\ elements of X intersects F. Note that the size of F is independent of the size of X. 1

On rainbow tetrahedra in Cayley graphs

Let $\Gamma_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $\Gamma_n$ are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs $G$ of largest degree 6, asymptotic diameter $|V(G)|^{1/3}$ and almost all vertices with degree: {\bf(a)} 6 in $G$; {\bf(b)} 4 in exactly six connected subgraphs of the $(3,6,3,6)$-semi-regular tessellation; and {\bf(c)} 3 in exactly four connected subgraphs of the $\{6,3\}$-regular hexagonal tessellation. These vertices have as closed neighborhoods the union (in a fixed way) of closed neighborhoods in the ten respective resulting tessellations. Generalizing asymptotic results are discussed as well.Comment: 21 pages, 7 figure

Overlap properties of geometric expanders

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n\rightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $\epsilon>0$, there exists $K=K(\epsilon,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $\mu$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $\epsilon$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$