5,764 research outputs found
Induced Ramsey-type theorems
We present a unified approach to proving Ramsey-type theorems for graphs with
a forbidden induced subgraph which can be used to extend and improve the
earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham,
and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by
Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's
regularity lemma, thereby giving much better bounds. The same approach can be
also used to show that pseudo-random graphs have strong induced Ramsey
properties. This leads to explicit constructions for upper bounds on various
induced Ramsey numbers.Comment: 30 page
On the minimum degree of minimal Ramsey graphs for multiple colours
A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every
r-colouring of the edges of G contains a monochromatic copy of H. The graph G
is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph
of G possesses this property. Let s_r(H) denote the smallest minimum degree of
G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter
s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed
that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of
s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) =
r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in
both r and k, and we determine s_r(K_3) up to a factor of log r
Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions
In their seminal work, Chattopadhyay and Zuckerman (STOC\u2716) constructed a two-source extractor with error epsilon for n-bit sources having min-entropy {polylog}(n/epsilon). Unfortunately, the construction\u27s running-time is {poly}(n/epsilon), which means that with polynomial-time constructions, only polynomially-small errors are possible. Our main result is a {poly}(n,log(1/epsilon))-time computable two-source condenser. For any k >= {polylog}(n/epsilon), our condenser transforms two independent (n,k)-sources to a distribution over m = k-O(log(1/epsilon)) bits that is epsilon-close to having min-entropy m - o(log(1/epsilon)). Hence, achieving entropy gap of o(log(1/epsilon)).
The bottleneck for obtaining low error in recent constructions of two-source extractors lies in the use of resilient functions. Informally, this is a function that receives input bits from r players with the property that the function\u27s output has small bias even if a bounded number of corrupted players feed adversarial inputs after seeing the inputs of the other players. The drawback of using resilient functions is that the error cannot be smaller than ln r/r. This, in return, forces the running time of the construction to be polynomial in 1/epsilon.
A key component in our construction is a variant of resilient functions which we call entropy-resilient functions. This variant can be seen as playing the above game for several rounds, each round outputting one bit. The goal of the corrupted players is to reduce, with as high probability as they can, the min-entropy accumulated throughout the rounds. We show that while the bias decreases only polynomially with the number of players in a one-round game, their success probability decreases exponentially in the entropy gap they are attempting to incur in a repeated game
Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs
In his 1947 paper that inaugurated the probabilistic method, Erd\H{o}s proved
the existence of -Ramsey graphs on vertices. Matching Erd\H{o}s'
result with a constructive proof is a central problem in combinatorics, that
has gained a significant attention in the literature. The state of the art
result was obtained in the celebrated paper by Barak, Rao, Shaltiel and
Wigderson [Ann. Math'12], who constructed a
-Ramsey graph, for some small universal
constant .
In this work, we significantly improve the result of Barak~\etal and
construct -Ramsey graphs, for some universal constant .
In the language of theoretical computer science, our work resolves the problem
of explicitly constructing two-source dispersers for polylogarithmic entropy
On some low distortion metric Ramsey problems
In this note, we consider the metric Ramsey problem for the normed spaces
l_p. Namely, given some 1=1, and an integer n, we ask
for the largest m such that every n-point metric space contains an m-point
subspace which embeds into l_p with distortion at most alpha. In
[arXiv:math.MG/0406353] it is shown that in the case of l_2, the dependence of
on alpha undergoes a phase transition at alpha=2. Here we consider this
problem for other l_p, and specifically the occurrence of a phase transition
for p other than 2. It is shown that a phase transition does occur at alpha=2
for every p in the interval [1,2]. For p>2 we are unable to determine the
answer, but estimates are provided for the possible location of such a phase
transition. We also study the analogous problem for isometric embedding and
show that for every 1<p<infinity there are arbitrarily large metric spaces, no
four points of which embed isometrically in l_p.Comment: 14 pages, to be published in Discrete and Computational Geometr
Designing Networks with Good Equilibria under Uncertainty
We consider the problem of designing network cost-sharing protocols with good
equilibria under uncertainty. The underlying game is a multicast game in a
rooted undirected graph with nonnegative edge costs. A set of k terminal
vertices or players need to establish connectivity with the root. The social
optimum is the Minimum Steiner Tree. We are interested in situations where the
designer has incomplete information about the input. We propose two different
models, the adversarial and the stochastic. In both models, the designer has
prior knowledge of the underlying metric but the requested subset of the
players is not known and is activated either in an adversarial manner
(adversarial model) or is drawn from a known probability distribution
(stochastic model).
In the adversarial model, the designer's goal is to choose a single,
universal protocol that has low Price of Anarchy (PoA) for all possible
requested subsets of players. The main question we address is: to what extent
can prior knowledge of the underlying metric help in the design? We first
demonstrate that there exist graphs (outerplanar) where knowledge of the
underlying metric can dramatically improve the performance of good network
design. Then, in our main technical result, we show that there exist graph
metrics, for which knowing the underlying metric does not help and any
universal protocol has PoA of , which is tight. We attack this
problem by developing new techniques that employ powerful tools from extremal
combinatorics, and more specifically Ramsey Theory in high dimensional
hypercubes.
Then we switch to the stochastic model, where each player is independently
activated. We show that there exists a randomized ordered protocol that
achieves constant PoA. By using standard derandomization techniques, we produce
a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu
Graphs with few 3-cliques and 3-anticliques are 3-universal
For given integers k, l we ask whether every large graph with a sufficiently
small number of k-cliques and k-anticliques must contain an induced copy of
every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A
similar phenomenon is established as well for tournaments with k=l=4.Comment: 12 pages, 1 figur
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