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106
Normgraphs: variations and applications
 J. Combin. Theory Ser. B
, 1999
"... We describe several variants of the normgraphs introduced by Kollár, Rónyai, and Szabó and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 1 2 n5/3 edges, containing no copy of K3,3, thus slightly impr ..."
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Cited by 41 (6 self)
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We describe several variants of the normgraphs introduced by Kollár, Rónyai, and Szabó and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 1 2 n5/3 edges, containing no copy of K3,3, thus slightly improving an old construction of Brown. We also prove that the maximum number of vertices in a complete graph whose edges can be colored by k colors with no monochromatic copy of K3,3 is (1 + o(1))k 3. This answers a question of Chung and Graham. In addition we prove that for every fixed t, there is a family of subsets of an n element set whose socalled dual shatter function is O(m t) and whose discrepancy is Ω(n 1/2−1/2t √ log n). This settles a problem of Matouˇsek.
Turán Numbers of Bipartite Graphs and Related RamseyType Questions
, 2003
"... For a graph H and an integer n, theTurán number ex(n, H) isthemaximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is rdegenerate if every one of its subgraphs contains a vertex of degree at most r. Weprove that, for any fixed bipartite graph H in which all ..."
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Cited by 30 (20 self)
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For a graph H and an integer n, theTurán number ex(n, H) isthemaximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is rdegenerate if every one of its subgraphs contains a vertex of degree at most r. Weprove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, ex(n, H) �O(n 2−1/r). This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite rdegenerate graph H, ex(n, H) �O(n 1−c/r). This is motivated by aconjectureofErdős that asserts that, for every such H, ex(n, H) �O(n 1−1/r). For two graphs G and H, the Ramsey number r(G, H) istheminimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either aredcopy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, r(G, G) �2 c √ m.Hereweprove this conjecture for bipartite graphs G, andprove that for general graphs G with m edges,
ON THE COMBINATORIAL PROBLEMS WHICH I WOULD MOST LIKE TO SEE SOLVED
, 1979
"... I was asked to write a paper about the major unsolved problems in combinatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no re ..."
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Cited by 23 (0 self)
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I was asked to write a paper about the major unsolved problems in combinatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no real competence but also topics about which I never seriously thought, e.g. algorithmic combinatorics, coding theory and matroid theory. There is no doubt that the proof of the conjecture that several simply stated problems have no good algorithm is fundamental and may have important consequences for many other branches of mathematics, but unfortunately I have no real feeling for these questions and I feel I should leave the subject to those who are more competent. I just heard that Khachiyan [59], has a polynomial algorithm for linear programming. (See also [50].) This is considered a sensational result and during my last stay in the U.S. many of my friends were greatly impressed by it.
Testing of Matrix Properties
 Proceedings of the 33 rd ACM STOC
, 2001
"... Combinatorial property testing, initiated by Rubinfeld and Sudan [15] and formally dened by Goldreich, Goldwasser and Ron in [12], deals with the following relaxation of decision problems: Given a xed property P and an input f , distinguish between the case that f satises P , and the case that no ..."
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Cited by 22 (9 self)
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Combinatorial property testing, initiated by Rubinfeld and Sudan [15] and formally dened by Goldreich, Goldwasser and Ron in [12], deals with the following relaxation of decision problems: Given a xed property P and an input f , distinguish between the case that f satises P , and the case that no input that diers from f in less than some xed fraction of the places satises P . An (; q)test for P is a randomized algorithm that queries at most q places of an input x and distinguishes with probability 2/3 between the case that f has the property and the case that at least an fraction of the places of f need to be changed in order for it to have the property. Here we concentrate on labeled, ddimensional grids, where the grid is viewed as a partially ordered set (poset) in the standard way (i.e as a product order of total orders). The main result here presents an (; poly(1=))test for every property of 0/1 labeled, ddimensional grids that is characterized by a nite collection of forbidden induced posets. Such properties include the `monotonicity' property studied in [7, 6], other more complicated forbidden chain patterns, and general forbidden poset patterns. We also present a (less ecient) test for such properties of labeled grids with larger xed size alphabets. All the above tests have in addition a 1sided error probability. Another result is a test for any bipartite graph property that is characterized by a nite set of forbidden induced subgraphs. Our test for such properties requires signicantly less queries than the previously known algorithm. Both collections above are variants of properties that are dened by certain rst order formulae with no quantier alternation over the syntax containing the grid order relations (and some add...
Blowup Lemma
 COMBINATORICA
, 1997
"... Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs. ..."
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Cited by 21 (2 self)
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Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs.
On Graphs which Contain All Small Trees
, 1978
"... We investigate those graphs G, with the property that any tree on N vertices occurs as subgraph of G,. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by $z log II and nl+l/log log %, respecti ..."
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Cited by 17 (4 self)
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We investigate those graphs G, with the property that any tree on N vertices occurs as subgraph of G,. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by $z log II and nl+l/log log %, respectively.
Extremal results for random discrete structures
"... Abstract. We study thresholds for extremal properties of random discrete structures. In particular, we determine the threshold for Szemerédi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Turántype proble ..."
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Cited by 17 (6 self)
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Abstract. We study thresholds for extremal properties of random discrete structures. In particular, we determine the threshold for Szemerédi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Turántype problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, Luczak, and Rödl for Turántype problems in random graphs. Similar results were obtained by Conlon and Gowers. 1.
Testing Satisfiability
, 2003
"... Let \Phi be a set of general logic functions on n variables... ..."
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Cited by 15 (7 self)
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Let \Phi be a set of general logic functions on n variables...