Problems and results in combinatorial analysis (1971)
| Venue: | COMBINATORICS (PROC. SYMP. PURE MATH |
| Citations: | 29 - 0 self |
BibTeX
@ARTICLE{Erdős71problemsand,
author = {P. Erdős},
title = {Problems and results in combinatorial analysis},
journal = {COMBINATORICS (PROC. SYMP. PURE MATH},
year = {1971},
pages = {77--89}
}
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Abstract
This review of some solved and unsolved problems in combinatorial analysis will be highly subjective. i will only discuss problems which I either worked on or at least thought about. The disadvantages of such an approach are obvious, but the disadvantages are perhaps counterbalanced by the fact that I certainly know more about these problems than about others (which perhaps are more important). i will mainly discuss finite combinatorial problems. I cannot claim completeness in any way but will try to refer to the literature in some cases; even so many things will be omitted. ISO will denote the cardinal number of S; c, cl, c2,... will denote absolute constants not necessarily the same at each occurrence. I. I will start with some problems dealing with subsets of a set. Let IS I =n. A well known theorem of Sperner [57] states that if A i a S, 15 i 5 m, is such that no A, contains any other, then max m=(aA). The theorem of Sperner has many applications in number theory; as far as I know these were first noticed by Behrend [2] and myself [8]. I asked 30 years ago several further extremal problems about subsets which also have number theoretic consequences. Let At a S, 15 i 5mi, assume that there are no three distinct A's so that Ai V A! = A,. I conjectured that
