209 research outputs found
Covering pairs by q2 + q + 1 sets
AbstractFor given k and s let n(k, s) be the largest cardinality of a set whose pairs can be covered by sk-sets. We determine n(k, q2 + q + 1) if a PG(2, q) exists, k > q(q + 1)2, and the remainder of k divided by (q + 1) is at least √q. Asymptotic results are also given for n(k, s) whenever s is fixed and k → ∞. Our main tool is the theory of fractional matchings of hypergraphs
Upper and Lower Bounds on Long Dual-Paths in Line Arrangements
Given a line arrangement with lines, we show that there exists a
path of length in the dual graph of formed by its
faces. This bound is tight up to lower order terms. For the bicolored version,
we describe an example of a line arrangement with blue and red lines
with no alternating path longer than . Further, we show that any line
arrangement with lines has a coloring such that it has an alternating path
of length . Our results also hold for pseudoline
arrangements.Comment: 19 page
Game saturation of intersecting families
We consider the following combinatorial game: two players, Fast and Slow,
claim -element subsets of alternately, one at each turn,
such that both players are allowed to pick sets that intersect all previously
claimed subsets. The game ends when there does not exist any unclaimed
-subset that meets all already claimed sets. The score of the game is the
number of sets claimed by the two players, the aim of Fast is to keep the score
as low as possible, while the aim of Slow is to postpone the game's end as long
as possible. The game saturation number is the score of the game when both
players play according to an optimal strategy. To be precise we have to
distinguish two cases depending on which player takes the first move. Let
and denote the score of
the saturation game when both players play according to an optimal strategy and
the game starts with Fast's or Slow's move, respectively. We prove that
holds
Linear Programming in the Semi-streaming Model with Application to the Maximum Matching Problem
In this paper, we study linear programming based approaches to the maximum
matching problem in the semi-streaming model. The semi-streaming model has
gained attention as a model for processing massive graphs as the importance of
such graphs has increased. This is a model where edges are streamed-in in an
adversarial order and we are allowed a space proportional to the number of
vertices in a graph.
In recent years, there has been several new results in this semi-streaming
model. However broad techniques such as linear programming have not been
adapted to this model. We present several techniques to adapt and optimize
linear programming based approaches in the semi-streaming model with an
application to the maximum matching problem. As a consequence, we improve
(almost) all previous results on this problem, and also prove new results on
interesting variants
The early evolution of the H-free process
The H-free process, for some fixed graph H, is the random graph process
defined by starting with an empty graph on n vertices and then adding edges one
at a time, chosen uniformly at random subject to the constraint that no H
subgraph is formed. Let G be the random maximal H-free graph obtained at the
end of the process. When H is strictly 2-balanced, we show that for some c>0,
with high probability as , the minimum degree in G is at least
. This gives new lower bounds for
the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite
graphs with . When H is a complete graph with we show that for some C>0, with high probability the independence number of
G is at most . This gives new lower bounds
for Ramsey numbers R(s,t) for fixed and t large. We also obtain new
bounds for the independence number of G for other graphs H, including the case
when H is a cycle. Our proofs use the differential equations method for random
graph processes to analyse the evolution of the process, and give further
information about the structure of the graphs obtained, including asymptotic
formulae for a broad class of subgraph extension variables.Comment: 36 page
The largest eigenvalue of rank one deformation of large Wigner matrices
The purpose of this paper is to establish universality of the fluctuations of
the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner
Ensembles. The real model is also considered. Our approach is close to the one
used by A. Soshnikov in the investigations of classical real or complex Wigner
Ensembles. It is based on the computation of moments of traces of high powers
of the random matrices under consideration
Random Tensors and Planted Cliques
The r-parity tensor of a graph is a generalization of the adjacency matrix,
where the tensor's entries denote the parity of the number of edges in
subgraphs induced by r distinct vertices. For r=2, it is the adjacency matrix
with 1's for edges and -1's for nonedges. It is well-known that the 2-norm of
the adjacency matrix of a random graph is O(\sqrt{n}). Here we show that the
2-norm of the r-parity tensor is at most f(r)\sqrt{n}\log^{O(r)}n, answering a
question of Frieze and Kannan who proved this for r=3. As a consequence, we get
a tight connection between the planted clique problem and the problem of
finding a vector that approximates the 2-norm of the r-parity tensor of a
random graph. Our proof method is based on an inductive application of
concentration of measure
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Tur\'an numbers for -free graphs: topological obstructions and algebraic constructions
We show that every hypersurface in contains a large grid,
i.e., the set of the form , with . We use this to
deduce that the known constructions of extremal -free and
-free graphs cannot be generalized to a similar construction of
-free graphs for any . We also give new constructions of
extremal -free graphs for large .Comment: Fixed a small mistake in the application of Proposition
Noise-Resilient Group Testing: Limitations and Constructions
We study combinatorial group testing schemes for learning -sparse Boolean
vectors using highly unreliable disjunctive measurements. We consider an
adversarial noise model that only limits the number of false observations, and
show that any noise-resilient scheme in this model can only approximately
reconstruct the sparse vector. On the positive side, we take this barrier to
our advantage and show that approximate reconstruction (within a satisfactory
degree of approximation) allows us to break the information theoretic lower
bound of that is known for exact reconstruction of
-sparse vectors of length via non-adaptive measurements, by a
multiplicative factor .
Specifically, we give simple randomized constructions of non-adaptive
measurement schemes, with measurements, that allow efficient
reconstruction of -sparse vectors up to false positives even in the
presence of false positives and false negatives within the
measurement outcomes, for any constant . We show that, information
theoretically, none of these parameters can be substantially improved without
dramatically affecting the others. Furthermore, we obtain several explicit
constructions, in particular one matching the randomized trade-off but using measurements. We also obtain explicit constructions
that allow fast reconstruction in time \poly(m), which would be sublinear in
for sufficiently sparse vectors. The main tool used in our construction is
the list-decoding view of randomness condensers and extractors.Comment: Full version. A preliminary summary of this work appears (under the
same title) in proceedings of the 17th International Symposium on
Fundamentals of Computation Theory (FCT 2009
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