Abstract
We consider an h-partite version of Dilworth’s theorem with multiple partial orders. Let P be a finite set, and let <1,...,< r be partial orders on P. Let G(P, <1,...,< r ) be the graph whose vertices are the elements of P, and x, y ∈ P are joined by an edge if x< i y or y< i x holds for some 1 ≤ i ≤ r. We show that if the edge density of G(P, <1, ... , < r ) is strictly larger than 1 − 1/(2h − 2)r, then P contains h disjoint sets A 1, ... , A h such that A 1 < j ... < j A h holds for some 1 ≤ j ≤ r, and |A 1| = ... = |A h | = Ω(|P|). Also, we show that if the complement of G(P, <) has edge density strictly larger than 1 − 1/(3h − 3), then P contains h disjoint sets A 1, ... , A h such that the elements of A i are incomparable with the elements of A j for 1 ≤ i < j ≤ h, and |A 1| = ... = |A h | = |P|1−o(1). Finally, we prove that if the edge density of the complement of G(P, <1, <2) is α, then there are disjoint sets A, B ⊂ P such that any element of A is incomparable with any element of B in both <1 and <2, and |A| = |B| > n 1−γ(α), where γ(α) → 0 as α → 1. We provide a few applications of these results in combinatorial geometry, as well.
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Tomon, I. Turán-Type Results for Complete h-Partite Graphs in Comparability and Incomparability Graphs. Order 33, 537–556 (2016). https://doi.org/10.1007/s11083-015-9384-6
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DOI: https://doi.org/10.1007/s11083-015-9384-6