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Grzesik A.,Jagiellonian University | Mikalacki M.,University of Novi Sad | Nagy Z.L.,Alfred Renyi Institute of Mathematics | Naor A.,Tel Aviv University | And 3 more authors.
Discrete Mathematics and Theoretical Computer Science | Year: 2015

In this paper, we study (1:b) Avoider-Enforcer games played on the edge set of the complete graph on n vertices. For every constant k ≥ 3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We consider both versions of Avoider-Enforcer games - the strict and the monotone - and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases fmonF, f-F and f +F, where jr is the hypergraph of the game. © 2015 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France. Source


Patkos B.,MTA ELTE Geometric and Algebraic Combinatorics Research Group | Patkos B.,Hungarian Academy of Sciences
Electronic Journal of Combinatorics | Year: 2015

The problem of determining the maximum size La(n, P) that a P-free subposet of the Boolean lattice Bn can have, attracted the attention of many researchers, but little is known about the induced version of these problems. In this paper we determine the asymptotic behavior of La*(n, P), the maximum size that an induced P-free subposet of the Boolean lattice Bn can have for the case when P is the complete two-level poset Kr,t or the complete multi-level poset Kr,s1,…,sj,t when all si's either equal 4 or are large enough and satisfy an extra condition. We also show lower and upper bounds for the non-induced problem in the case when P is the complete three-level poset Kr,s,t. These bounds determine the asymptotics of La(n, Kr,s,t) for some values of s independently of the values of r and t. © 2015, International Press of Boston, Inc. All rights reserved. Source


Barat J.,Monash University | Barat J.,MTA ELTE Geometric and Algebraic Combinatorics Research Group | Gerbner D.,Hungarian Academy of Sciences
Electronic Journal of Combinatorics | Year: 2014

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 kT such that if G is a kT -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. The same result has been independently obtained by Carsten Thomassen (2013). 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. Recently Carsten Thomassen proved a more general decomposition theorem for bistars, which yields the same result with a worse constant. Keywords: graph theory; decomposition; tree; edge-connectivity. Source


Fancsali S.L.,MTA ELTE Geometric and Algebraic Combinatorics Research Group | Sziklai P.,Institute of Mathematics
Electronic Journal of Combinatorics | Year: 2014

In this article, we examine sets of lines in PG(d, F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least [1.5d] lines if the field F has at least [1.5d] elements, and at least 2d - 1 lines if the field F is algebraically closed. We show that suitable 2d - 1 lines constitute such a set (if |F| ≥ 2d - 1), proving that the lower bound is tight over algebraically closed fields. At last, we will see that the strong (s, A) subspace designs constructed by Guruswami and Kopparty have better (smaller) parameter A than one would think at first sight. Source


Heger T.,MTA ELTE Geometric and Algebraic Combinatorics Research Group | Patkos B.,MTA ELTE Geometric and Algebraic Combinatorics Research Group | Patkos B.,Hungarian Academy of Sciences | Takats M.,Eotvos Lorand University
Designs, Codes, and Cryptography | Year: 2015

We consider the following q-analog of the basic combinatorial search problem: let q be a prime power and GF(q) the finite field of q elements. Let V denote an n-dimensional vector space over GF(q) and let v be an unknown 1-dimensional subspace of V. We will be interested in determining the minimum number of queries that is needed to find v provided all queries are subspaces of V and the answer to a query U is YES if v⩽U and NO if v⩽̸U. This number will be denoted by A(n,q) in the adaptive case (when for each queries answers are obtained immediately and later queries might depend on previous answers) and M(n,q) in the non-adaptive case (when all queries must be made in advance). In the case n=3 we prove 2q-1=A(3,q) Source

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