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The Regularity Lemma of Szemer edi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we rst demonstrate the computational diiculty of nding a regular partition; we show that deciding if a given partition of an input graph satisses the… (More)

We consider anti-Ramsey type results. For a given coloring ∆ of the k-element subsets of an n-element set X, where two k-element sets with nonempty intersection are colored differently, let inj ∆ (k, n) be the largest size of a subset Y ⊆ X, such that the k-element subsets of Y are colored pairwise differently. Taking the minimum over all colorings, i.e.… (More)

In this paper typical properties of large random Boolean AND/OR formulas are investigated. Such formulas with n variables are viewed as rooted binary trees chosen from the uniform distribution of all rooted binary trees on m nodes, where n is xed and m tends to innnity. The leaves are labeled by literals and the inner nodes by the connectives AND/OR, both… (More)

In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by ErdSs and Rado, are given. These yield improvements over the known bounds for the arising Erd6s-Rado numbers ER(k; l), where the numbers ER(k; l) are defined as the least positive integer n such that for every partition of the k-element subsets of a totally ordered… (More)

We consider the problem of finding deterministically a large independent set of guaranteed size in a hyper-graph on n vertices and with m edges. With respect to the Tur&n bound, the quality of our solutions is for hypergraphs with not too many small cycles by a logarithmic factor in the input size better. The algorithms are fast. Namely, they often have a… (More)

Heilbronn conjectured that given arbitrary n points from the 2-dimensional unit square, there must be three points which form a triangle of area at most O(1/n2). This conjecture was disproved by a nonconstructive argument of Komlds, Pintz and Szemer~di [7] who showed that for every n there is a configuration of n points in the unit square where all… (More)

For positive integers d and n let fd(n) denote the maximum cardinality of a subset of the n d-grid f1; 2; : : : ; ng d with distinct mutual euclidean distances. Improving earlier results of Erd} os and Guy, it will be shown that f2(n) c n 2=3 and, for d 3, that fd(n) cd n 2=3 (ln n) 1=3 , where c; cd > 0 are constants. Also improvements of lower bounds of… (More)