Some problems in combinatorial analysis

Some problems in combinatorial analysis

Combinatorial analysis

Abbott, Harvey Leslie

1965

Moser, Leo, 1921-1970

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This thesis is devoted to a study of a variety of problems in combinatorial analysis. In Chapter I some new results on a problem of Schur concerning sum-free sets of integers are obtained. Denote by f(n) the largest positive integer m for which there exists some way of partitioning the integers 1,2, … , m into n sum free sets. It is proved that f(n) > 89 n/4 - c log n for some absolute constant c and all sufficiently large n. The methods developed can be applied successfully to other related questions. In chapter II some recurrence inequalities for certain of the Ramsay numbers are obtained and some new lower bounds for these numbers are derived. In chapter III we investigate finite families F of finite sets which possess the following property: if B F is such that B ∩ F ≠ Φ for each F F, the B F for some F F. Two questions of Erdös and Hajnal are considered, and a special case of Ramsay’s Theorem is used to settle one of the questions. In Chapter IV some improvements on a combinatorial theorem of Erdös and Rado are obtained and an application of the theorem to a problem in number theory is discussed. Finally, in Chapter V, we consider the construction and enumeration of certain types of paths on the n-dimensional unit cube. In particular, a new lower bound for the number of Hamiltonian cycles on the n-cube is obtained.

2015-06-01

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