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The Schur degree of additive sets. (English) Zbl 1472.11079

Summary: Let \(( G,+)\) be an abelian group. A subset of \(G\) is sumfree if it contains no elements \(x,y,z\) such that \(x + y = z\). We extend this concept by introducing the Schur degree of a subset of \(G\), where Schur degree 1 corresponds to sumfree. The classical inequality \(S(n) \leq R_n ( 3 ) - 2\), between the Schur number \(S(n)\) and the Ramsey number \(R_n(3) = R( 3, \ldots, 3)\), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of \(G\). Recursive upper bounds are known for \(R_n(3)\) but not for \(S(n)\) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our study of the Schur degree leads us to conjecture \(S(n) \leq n(S(n - 1) + 1)\) for all \(n \geq 2\). If true, it would yield substantially better upper bounds on the Schur numbers, e.g. \(S(6) \leq 966\) conjecturally, whereas all is known so far is \(536 \leq S(6) \leq 1836\).

MSC:

11B75 Other combinatorial number theory
05D10 Ramsey theory

Software:

Mathematica
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Full Text: DOI arXiv

References:

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