Measures of sum-free intersecting families
The paper originates from \textit{V. Bergelson}'s search for a density statement of Schur's theorem in Ramsey theory concerning any finite partitioning of the set of positive integers, resulting in an ergodic-theoretical presentation [J. Comb. Theory, Ser. A 43, 338--343 (1986; Zbl 0607.10040)]. It is also informed by a problem in combinatorics concerning sets of integers containing terms in arithmetic progression. Let \(\alpha\) be the supremum of all the \(\delta\) such that there exists a sequence \(\langle A_n\rangle_{n=1}^\infty\) of measurable subsets of \((0, 1)\) with the property that the Lebesgue measures \(\mu (A_n) \geq \delta\) for all \(n, m \in N\), \(A_n\cap A_m \cap A_{n+m} = \emptyset \); for \(k \in N\), \(\alpha_k\) the corresponding supremum for \(\langle A_n\rangle_{n=1}^k\). This paper shows that \( \alpha = \lim_{k \to \infty} \alpha_{k}\), and that \(\alpha_{k}= \frac{4}{9}\) for \(k = 5, 6, 7\). Using Pascal programs and the Mathematica Linear Programming program, the authors computed the exact values of \(\alpha_{k}\) for \(k \leq 41\), as well as, for \(k \leq 41\), the exact number of maximal sum-free subsets of \(\{ 1, \ldots, k\}\) (for any two elements in such a subset, the sum is not in it and the subset is not a proper subset of any such subsets). The paper also investigates the size of the sets \(\langle A_{x}\rangle_{x \in S}\) with \(A_{x} \cap A_{y} \cap A_{x+y} = \emptyset \), where \(S\) is a subsemigroup of \(((0, \infty), +)\), showing for example that \(\{ A_{x}\}_{x \in Q^+} \) can be found such that each \(\mu (A_{x})= \frac{1}{3}\).
scientific article; zbMATH DE number 5137997