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Explicit Ramsey graphs and orthonormal labelings

Noga Alon
1994 Electronic Journal of Combinatorics  
We describe an explicit construction of triangle-free graphs with no independent sets of size $m$ and with $\Omega(m^{3/2})$ vertices, improving a sequence of previous constructions by various authors.  ...  Kashin and Konyagin who showed that this maximum is at least $\Omega(n^{1/3}/ \log n)$ and at most $O(n^{1/3})$.  ...  Acknowledgment I would like to thank Nabil Kahale for helpful comments and Rob Calderbank for fruitful suggestions that improved the presentation significantly.  ... 
doi:10.37236/1192 fatcat:vmjowc6khjhu3mnmwuvr4xakhe

Approximating the independence number via theϑ-function

Noga Alon, Nabil Kahale
1998 Mathematical programming  
and the independence number of a graph on n vertices.  ...  The algorithm is based on semidefinite programming, some properties of the Lovász ϑ-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a  ...  Acknowledgement We would like to thank Mario Szegedy for helpful discussions, suggestions and comments.  ... 
doi:10.1007/bf01581168 fatcat:lvnnqsbqtrfwxnzbggnpahethq

Page 6439 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews  
Several types of construction algorithms of S¢- factorization of K,,,, are also given.” 95k:05145 05C75 0sC55 Alon, Noga (1-BELL; Murray Hill, NJ) Explicit Ramsey graphs and orthonormal labelings.  ...  The author describes an explicit construction of triangle-free graphs with no independent set of order m and with Q(m?/) vertices.  ... 

Randomness and Pseudo-Randomness in Discrete Mathematics [chapter]

Noga Alon
1998 PROGRESS IN MATHEMATICS  
To this end, certain explicit pseudo-random structures are needed, and their construction often requires tools from a wide variety of mathematical areas including Group Theory, Number Theory and Algebraic  ...  In this paper I survey some of these developments and mention several related open problems.  ...  The best known problem of finding an explicit construction of a combinatorial structure is probably that of constructing explicit Ramsey graphs.  ... 
doi:10.1007/978-3-0348-8974-2_1 fatcat:j6bavkgixvbspapsnbt47hf75m

On the Global Self-attention Mechanism for Graph Convolutional Networks [article]

Chen Wang, Chengyuan Deng
2020 arXiv   pre-print
However, it is not clear if Graph Convolutional Networks (GCNs) can similarly benefit from such a technique.  ...  and the theoretical results.  ...  The loss function can be denoted as: L(ÃH (L) + γBH (L) , y), (5) where y is the target (the labels of nodes or graph).  ... 
arXiv:2010.10711v1 fatcat:qg2yijlvcjczfgqiglckhfdp2y

Complexity Lower Bounds using Linear Algebra

Satyanarayana V. Lokam
2007 Foundations and Trends® in Theoretical Computer Science  
Sharma, Madhu Sudan, and Ramarathnam Venkatesan.  ...  Acknowledgments I am grateful to the following people for their careful reading of the early versions of this paper and valuable feedback: Amit Deshpande, Seny Kamara, Neeraj Kayal, Meena Mahajan, Jayalal  ...  Thus, a graph formula can be represented as a tree with the leaves labeled by generator graphs and the internal nodes labeled by operations from the basis.  ... 
doi:10.1561/0400000011 fatcat:jso5gaqnhnbitierw4ytzhixna

Page 151 of Mathematical Reviews Vol. 27, Issue Index [page]

Mathematical Reviews  
(English summary) 95k:05127 Zhang, Xiao Xian see Xie, Ji Guo, 95b:05152 Items with secondary classifications in 05C55 Alon, Noga Explicit Ramsey graphs and orthonormal labelings.  ...  Ramsey size linear graphs. (English summary) 95¢:05087 Exoo, Geoffrey A lower bound for Schur numbers and multicolor Ramsey numbers of K3.  ... 

How (Maximally) Contextual is Quantum Mechanics? [article]

Andrew W. Simmons
2017 arXiv   pre-print
this paper, we demonstrate a bound on what fraction of rank-1 projective measurements on a quantum system must be considered to have context-dependent valuations as a function of the quantum dimension, and  ...  The proof proceeds by demonstrating an explicit procedure for finding a T and an A ⊂ T , for any {|ψ i }, with upper-bounded worst-case performance.  ...  This reduction of our figure-of-merit allows us to make use of a bound derived for Ramsey type problems.  ... 
arXiv:1712.03766v1 fatcat:zdg3iykjunajzmatwpnvucc5lu

Frames, graphs and erasures

Bernhard G. Bodmann, Vern I. Paulsen
2005 Linear Algebra and its Applications  
Conversely, given an n × n self-adjoint projection P of rank k, we can always factor it as P = V V * with an n × k matrix V , by choosing an orthonormal basis for the range of P as the column vectors of  ...  If V and W are the analysis operators for F and G, respectively, then it is clear that F and G are type I equivalent if and only if V = W U or equivalently, if and only if V V * = W W * .  ...  Acknowledgments This research was partially supported by the following grants: NSF grant DMS-0300128, University of Houston TLCC funds, and the Texas Higher Education Coordinating Board Grant ARP-003652  ... 
doi:10.1016/j.laa.2005.02.016 fatcat:jgvcwds3gvbgxhp32ittkfjhhu

Frames, Graphs and Erasures [article]

Bernhard G. Bodmann, Vern I. Paulsen
2005 arXiv   pre-print
Two-uniform frames and their use for the coding of vectors are the main subject of this paper.  ...  We derive general error bounds for two-uniform frames when more than two erasures occur and apply these to concrete examples.  ...  This research was partially supported by the following grants: NSF grant DMS-0300128, University of Houston TLCC funds, and the Texas Higher Education Coordinating Board Grant ARP-003652-0071-2001.  ... 
arXiv:math/0406134v3 fatcat:lhjvb6ytkzg35ofeteab42eyqy

The length of an s-increasing sequence of r-tuples [article]

W. T. Gowers, J. Long
2016 arXiv   pre-print
We prove a number of results related to a problem of Po-Shen Loh, which is equivalent to a problem in Ramsey theory. Let a=(a_1,a_2,a_3) and b=(b_1,b_2,b_3) be two triples of integers.  ...  We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well known problems in extremal combinatorics, and asking a number of  ...  It is easiest to visualise the graph G as a labelled bipartite graph G ′ on X × Y , where the label on the edge (x, y) is z (for each face (x, y, z) of G).  ... 
arXiv:1609.08688v2 fatcat:tif7h2xwdffp7iui5eeb3xtvda

Mean-field dynamics of two-mode Bose–Einstein condensates in highly anisotropic potentials: interference, dimensionality and entanglement

Alexandre B Tacla, Carlton M Caves
2013 New Journal of Physics  
We study the relative-phase dynamics in the context of a nonlinear Ramsey interferometry scheme, which has recently been proposed as a novel platform for high-precision interferometry.  ...  We use this model to investigate the relative phase and density dynamics of a two-mode, cigar-shaped ^87Rb BEC.  ...  PHY-0903953 and PHY-1005540.  ... 
doi:10.1088/1367-2630/15/2/023008 fatcat:j3cutvsgdvfhrdadoa7j7wfz2a

Additive Combinatorics and its Applications in Theoretical Computer Science

Shachar Lovett
2017 Theory of Computing  
extraction and pseudo-randomness.  ...  Define A + B = {a + b : a ∈ A, b ∈ B} and A − B = {a − b : a ∈ A, b ∈ B} to be their sumset and difference set, respectively.  ...  Both Ramsey graphs and two-source extractors have been extensively studied in the graph theory and computational complexity communities.  ... 
doi:10.4086/toc.gs.2017.008 dblp:journals/toc/Lovett17 fatcat:z5d45zpq3vgpzk7tshdszrdwni

Does Productivity Affect Unemployment? A Time-Frequency Analysis for the US [chapter]

Marco Gallegati, Mauro Gallegati, James B. Ramsey, Willi Semmler
2014 Dynamic Modeling and Econometrics in Economics and Finance  
We decompose the relevant US time series data in different time scale components and consider co-movements of productivity and unemployment over different time horizons.  ...  Productivity growth is recognized as a major force to increase the overall performance of the economy, as measured for example by the growth of output, real wages, and cost reduction, and a major source  ...  We thank all participants for valuable comments and suggestions, particularly Jouko Vilmunen and Patrick Crowley.  ... 
doi:10.1007/978-3-319-07061-2_2 fatcat:aubnchcggrdk5arkdwosdpd6lu

Small Ramsey Numbers

Stanisław Radziszowski
unpublished
We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete  ...  We give references to all cited bounds and values, as well as to previous similar compilations.  ...  Acknowledgement I would like to thank Brendan McKay, Geoffrey Exoo and Heiko Harborth for their help in gathering data for earlier versions of this survey.  ... 
fatcat:qbqs3msi2zccxkmlriie2rjej4
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