arXiv subjects

Dániel Gerbner

Publications and source records attributed to Dániel Gerbner.

121 records · Page 7Linked to original sources

arXiv2016

A note on tilted Sperner families with patterns

Let $p$ and $q$ be two nonnegative integers with $p+q>0$ and $n>0$. We call $\mathcal{F} \subset \mathcal{P}([n])$ a \textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \in \mathcal{F}$ with: $$(i) \ \ p|F \setminus G|=q|G \setminus F|, \ \textrm{and}$$ $$(ii) \ f > g \ \textrm{for all} \ f \in F \setminus G \ \textrm{and} \ g \in G \setminus F.$$ Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\sqrt{\log n}}\ \frac{2^n}{\sqrt{n}}).$$ We improve and generalize this result, and prove that the cardinality of every ($p,q$)-tilted Sperner family with patterns on [$n$] is $$O(\sqrt{\log n} \ \frac{2^n}{\sqrt{n}}).$$

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arXiv2015

Generalising separating families of fixed size

We examine the following version of a classic combinatorial search problem introduced by Rényi: Given a finite set $X$ of $n$ elements we want to identify an unknown subset $Y \subset X$ of exactly $d$ elements by testing, by as few as possible subsets $A$ of $X$, whether $A$ contains an element of $Y$ or not. We are primarily concerned with the model where the family of test sets is specified in advance (non-adaptive) and each test set is of size at most a given $k$. Our main results are asymptotically sharp bounds on the minimum number of tests necessary for fixed $d$ and $k$ and for $n$ tending to infinity.

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arXiv2015

Extremal results for Berge-hypergraphs

Let $G$ be a graph and $\mathcal{H}$ be a hypergraph both on the same vertex set. We say that a hypergraph $\mathcal{H}$ is a \emph{Berge}-$G$ if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for $e \in E(G)$ we have $e \subset f(e)$. This generalizes the established definitions of "Berge path" and "Berge cycle" to general graphs. For a fixed graph $G$ we examine the maximum possible size (i.e.\ the sum of the cardinality of each edge) of a hypergraph with no Berge-$G$ as a subhypergraph. In the present paper we prove general bounds for this maximum when $G$ is an arbitrary graph. We also consider the specific case when $G$ is a complete bipartite graph and prove an analogue of the K\H ovári-Sós-Turán theorem.

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arXiv2014

A discrete isodiametric result: the Erdős-Ko-Rado theorem for multisets

There are many generalizations of the Erdős-Ko-Rado theorem. We give new results (and problems) concerning families of $t$-intersecting $k$-element multisets of an $n$-set and point out connections to coding theory and classical geometry. We establish the conjecture that for $n \geq t(k-t)+2$ such a family can have at most ${n+k-t-1\choose k-t}$ members.

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arXiv2013

Advantage in the discrete Voronoi game

We study the discrete Voronoi game, where two players alternately claim vertices of a graph for t rounds. In the end, the remaining vertices are divided such that each player receives the vertices that are closer to his or her claimed vertices. We prove that there are graphs for which the second player gets almost all vertices in this game, but this is not possible for bounded-degree graphs. For trees, the first player can get at least one quarter of the vertices, and we give examples where she can get only little more than one third of them. We make some general observations, relating the result with many rounds to the result for the one-round game on the same graph.

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arXiv2012

Density-based group testing

In this paper we study a new, generalized version of the well-known group testing problem. In the classical model of group testing we are given n objects, some of which are considered to be defective. We can test certain subsets of the objects whether they contain at least one defective element. The goal is usually to find all defectives using as few tests as possible. In our model the presence of defective elements in a test set Q can be recognized if and only if their number is large enough compared to the size of Q. More precisely for a test Q the answer is 'yes' if and only if there are at least α|Q| defective elements in Q for some fixed α.

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arXiv2012

Edge-decomposition of graphs into copies of a tree with four edges

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a 191-edge-connected graph of size divisible by 4, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star.

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arXiv2012

Majority and Plurality Problems

Given a set of n balls each colored with a color, a ball is said to be majority, k-majority, plurality if its color class has size larger than half of the number of balls, has size at least k, has size larger than any other color class; respectively. We address the problem of finding the minimum number of queries (a comparison of a pair of balls if they have the same color or not) that is needed to decide whether a majority, k-majority or plurality ball exists and if so then show one such ball. We consider both adaptive and non-adaptive strategies and in certain cases, we also address weighted versions of the problems.

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arXiv2011

Generalizations of the Tree Packing Conjecture

The Gyárfás tree packing conjecture asserts that any set of trees with $2,3, ..., k$ vertices has an (edge-disjoint) packing into the complete graph on $k$ vertices. Gyárfás and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into $k$-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any $k$-chromatic graph. We also consider several other generalizations of the conjecture.

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arXiv2011

Two-part set systems

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\cF$ is a family of subsets of $X$ such that no two sets $A, B \in \cF$ satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B \cap X_i$ for some $i$, then $|\cF| \le {n \choose \lfloor n/2 \rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfiy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following new result which may be of independent interest: let $\cF, \cG$ be families of subsets of an $n$-element set such that $\cF$ and $\cG$ are both intersecting and cross-Sperner, meaning that if $A \in \cF$ and $B \in \cG$, then $A \not\subset B$ and $B \not\subset A$. Then $|\cF| +|\cG| < 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.

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arXiv2011

Saturating Sperner families

A family $\cF \subseteq 2^{[n]}$ saturates the monotone decreasing property $\cP$ if $\cF$ satisfies $\cP$ and one cannot add any set to $\cF$ such that property $\cP$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the $k$-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of $l$-sets and $(l+1)$-sets.

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arXiv2011

Path-search in the pyramid and in other graphs

We are given an acyclic directed graph with one source, and a subset of its edges which contains exactly one outgoing edge for every non-sink vertex. These edges determine a unique path from the source to a sink. We can think of it as a switch in every vertex, which determines which way the water arriving to that vertex flows further. We are interested in determining either the sink the flow arrives, or the whole path, with as few questions as possible. The questions we can ask correspond to the vertices of the graph, and the answer describes the switch, i.e. tells which outgoing edge is in our given subset. Originally the problem was proposed by Soren Riis (who posed the question for pyramid graphs) in the following more general form. We are given a natural number k, and k questions can be asked in a round. The goal is to minimize the number of rounds. We completely solve this problem for complete t-ary trees. Also, for pyramid graphs we present some non-trivial partial results.

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arXiv2011

Cross-Sperner families

A pair of families $(\cF,\cG)$ is said to be \emph{cross-Sperner} if there exists no pair of sets $F \in \cF, G \in \cG$ with $F \subseteq G$ or $G \subseteq F$. There are two ways to measure the size of the pair $(\cF,\cG)$: with the sum $|\cF|+|\cG|$ or with the product $|\cF|\cdot |\cG|$. We show that if $\cF, \cG \subseteq 2^{[n]}$, then $|\cF||\cG| \le 2^{2n-4}$ and $|\cF|+|\cG|$ is maximal if $\cF$ or $\cG$ consists of exactly one set of size $\lceil n/2 \rceil$ provided the size of the ground set $n$ is large enough and both $\cF$ and $\cG$ are non-empty.

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