Results 1  10
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31
A classification of rapidly growing Ramsey functions
 PROC. AMER. MATH. SOC
, 2003
"... Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf i ..."
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Cited by 16 (5 self)
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Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf is independent of first order (Peano) arithmetic PA.Iffis a fixed iteration of the binary length function, then PHf is independent. On the other hand PHlog ∗ is provable in PA. More precisely let fα(i):=i  H −1 α (i) where  i h denotes the htimes iterated binary length of i and H−1 α denotes the inverse function of the αth member Hα of the Hardy hierarchy. Then PHfα is independent of PA (for α ≤ ε0) iffα = ε0.
Approximating maximum independent sets in uniform hypergraphs
 In Proc. 23rd Intl. Symp
, 1998
"... We consider the problem of approximating the independence number and the chromatic number of kuniform hypergraphs on n vertices. For xed integers k 2, we obtain for both problems that one can achieve in polynomial time approximation ratios of at most O(n=(log (k,1) n) 2). This extends results of Bo ..."
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Cited by 15 (0 self)
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We consider the problem of approximating the independence number and the chromatic number of kuniform hypergraphs on n vertices. For xed integers k 2, we obtain for both problems that one can achieve in polynomial time approximation ratios of at most O(n=(log (k,1) n) 2). This extends results of Boppana and Halldorsson [5] who showed for the graph case that an approximation ratio of O(n=(log n) 2) can be achieved in polynomial time. On the other hand, assuming NP 6 = ZPP, one cannot obtain in polynomial time for the independence number and the chromatic number of kuniform hypergraphs an approximation ratio of n 1, for xed>0. 1
D.: Embeddings and Ramsey numbers of sparse kuniform hypergraphs, Combinatorica 29
, 2009
"... Abstract. Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [5, 19] the same result was proved for 3uniform hypergraphs. Here we extend this result to kuniform hypergraphs for any integer k ≥ 3. As in the 3uni ..."
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Cited by 13 (3 self)
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Abstract. Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [5, 19] the same result was proved for 3uniform hypergraphs. Here we extend this result to kuniform hypergraphs for any integer k ≥ 3. As in the 3uniform case, the main new tool which we prove and use is an embedding lemma for kuniform hypergraphs of bounded maximum degree into suitable kuniform ‘quasirandom ’ hypergraphs.
A new method to construct lower bounds for van der Waerden numbers. The Electronic
 Journal of Combinatorics
"... Supported by the Dutch Organization for Scientific Research (NWO) under grant 617.023.306 the electronic journal of combinatorics 14 (2007), #R6 1 1 Introduction In 1927 the Dutch mathematician Van der Waerden proved [18] a (generalization of) a conjecture of Schur ..."
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Cited by 13 (3 self)
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Supported by the Dutch Organization for Scientific Research (NWO) under grant 617.023.306 the electronic journal of combinatorics 14 (2007), #R6 1 1 Introduction In 1927 the Dutch mathematician Van der Waerden proved [18] a (generalization of) a conjecture of Schur
Satisfiability and computing van der Waerden numbers
, 2004
"... In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area o ..."
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Cited by 7 (3 self)
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In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden numbers as an example, we show that these problems can be represented by parameterized propositional theories in such a way that decisions concerning their satisfiability determine the numbers (function) in question. We show that by using generalpurpose complete and localsearch techniques for testing propositional satisfiability, this approach becomes e#ective  competitive with specialized approaches. By following it, we were able to obtain several new results pertaining to the problem of computing van der Waerden numbers. We also note that due to their properties, especially their structural simplicity and computational hardness, propositional theories that arise in this research can be of use in development, testing and benchmarking of SAT solvers.
Lattice Structures and Spreading Models
, 2007
"... We consider problems concerning the partial order structure of the set of spreading models of Banach spaces. We construct examples of spaces showing that the possible structure of these sets include certain classes of finite semilattices and countable lattices and all finite lattices. ..."
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Cited by 6 (3 self)
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We consider problems concerning the partial order structure of the set of spreading models of Banach spaces. We construct examples of spaces showing that the possible structure of these sets include certain classes of finite semilattices and countable lattices and all finite lattices.
Transforming comparison model lower bounds to the parallelrandomaccessmachine
 INFORMATION PROCESSING LETTERS
, 1997
"... We provide general transformations of lower bounds in Valiant's parallelcomparisondecisiontree model to lower bounds in the priority concurrentread concurrentwrite parallelrandomaccessmachine model. The proofs rely on standard Ramseytheoretic arguments that simplify the structure of the com ..."
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Cited by 5 (0 self)
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We provide general transformations of lower bounds in Valiant's parallelcomparisondecisiontree model to lower bounds in the priority concurrentread concurrentwrite parallelrandomaccessmachine model. The proofs rely on standard Ramseytheoretic arguments that simplify the structure of the computation by restricting the input domain. The transformation of comparison model lower bounds, which are usually easier to obtain, to the parallelrandomaccessmachine, unifies some known lower bounds and gives new lower bounds for several problems.
Hypergraph Ramsey numbers
"... The Ramsey number rk(s, n) is the minimum N such that every redblue coloring of the ktuples of an Nelement set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all ktuples from this set are red (blue). In this paper we obtain new estimates for seve ..."
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Cited by 5 (1 self)
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The Ramsey number rk(s, n) is the minimum N such that every redblue coloring of the ktuples of an Nelement set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all ktuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for rk(s, n) for k ≥ 3 and s fixed. In particular, we show that r3(s, n) ≤ 2 ns−2 log n, which improves by a factor of n s−2 /polylogn the exponent of the previous upper bound of Erdős and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there are constants c1, c2> 0 such that c1 sn log(n/s) r3(s, n) ≥ 2 for all 4 ≤ s ≤ c2n. When s is a constant, it gives the first superexponential lower bound for r3(s, n), answering an open question posed by Erdős and Hajnal in 1972. Next, we consider the 3color Ramsey number r3(n, n, n), which is the minimum N such that every 3coloring of the triples of an Nelement set contains a monochromatic set of size n. Improving another old result of Erdős and Hajnal, we show that n nclog r3(n, n, n) ≥ 2. Finally, we make some progress on related hypergraph Ramseytype problems. 1