We consider the problem of approximating the independence number and the chromatic number of k-uniform 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 k-uniform hypergraphs an approximation ratio of n 1, for xed>0. 1

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 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs for any integer k ≥ 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform ‘quasi-random ’ hypergraphs.

Let f be a number-theoretic function. A finite set X of natural numbers is called f-large if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding f-largeness 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 h-times iterated binary length of i and H−1 α denotes the inverse function of the α-th member Hα of the Hardy hierarchy. Then PHfα is independent of PA (for α ≤ ε0) iffα = ε0.

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

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 general-purpose complete and local-search 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.

We provide general transformations of lower bounds in Valiant's parallel-comparison-decision-tree model to lower bounds in the priority concurrent-read concurrent-write parallel-random-access-machine model. The proofs rely on standard Ramsey-theoretic 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 parallel-random-access-machine, unifies some known lower bounds and gives new lower bounds for several problems.

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 semi-lattices and countable lattices and all finite lattices.

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The Ramsey number rk(s, n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element 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 k-tuples 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 3-coloring of the triples of an N-element 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 Ramsey-type problems. 1