Arithmetic progressions in partially ordered sets
Van der Waerden's famous arithmetic progression theorem is generalized to partially ordered sets. As a model of thinking the authors use its `density version' by \textit{E. Szemerédi} [Acta Arith. 27, 199-245 (1975; Zbl 0303.10056)]: for every \(\epsilon >0\) and every positive integer t there exists \(N=N(\epsilon,t)\) so that for any \(n\geq N\) every subset \(S\subset \{1,2,...,n\}\) with \(| S| \geq \epsilon n\) contains a t- term arithmetic progression. A sequence of distinct elements \(a_ 1,a_ 2,...,a_ t\) of a poset P is said to be a t-term arithmetic progression in P if there exists natural d so that the number of elements in each interval \([a_ i,a_{i+1}]=\{x\in P|\) \(a_ i\leq x\leq a_{i+1}\}\), \(i\in \{1,2,...,t-1\}\) is d. The width of a (finite) poset P is the maximal number of pairwise incomparable elements in P. The authors prove (Theorem 3): For any \(\epsilon >0\) and any pair t, \(\omega\) of positive integers there exists \(N=N(\epsilon,t,\omega)\) so that in any poset with cardinality \(\geq N\) and with width(P)\(\leq \omega\) every subset \(Q\subseteq P\) with \(| Q| \geq \epsilon \cdot | P|\) contains a t-term arithmetic progression. In the concluding section of this interesting paper the following question is raised: is it true, that for every \(\epsilon >0\) and every positive integer t there exists \(M=M(\epsilon,t)\) so that in any distributive lattice L of cardinality \(\geq M\) every subset \(S\subset L\), \(| S| \geq \epsilon \cdot | L|\) contains a t-term arithmetic progression.
scientific article; zbMATH DE number 4023346