5 research outputs found
A multipartite analogue of Dilworth's Theorem
We prove that every partially ordered set on elements contains
subsets such that either each of these subsets has
size and, for every , every element in is less
than or equal to every element in , or each of these subsets has size
and, for every , every element in
is incomparable with every element in for . This answers a
question of the first author from 2006. As a corollary, we prove for each
positive integer there is such that for any partial orders
on a set of elements, there exists subsets
each of size at least such that
for each partial order , either
for any tuple of elements
, or
for any , or is incomparable with for
any , and . This improves on a 2009 result of
Pach and the first author motivated by problems in discrete geometry
Extremal Graph Theory and Dimension Theory for Partial Orders
This dissertation analyses several problems in extremal combinatorics.In Part I, we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph H and an integer t, what is the minimum number of coloured edges in a t-edge-coloured graph G on n vertices such that G does not contain a rainbow copy of H, but adding a new edge to G in any colour creates a rainbow copy of H? We determine the growth rates of these numbers for almost all graphs H and all t e(H).In Part II, we study dimension theory for finite partial orders. In Chapter 1, we introduce and define the concepts we use in the succeeding chapters.In Chapter 2, we determine the dimension of the divisibility order on [n] up to a factor of (log log n).In Chapter 3, we answer a question of Kim, Martin, Masak, Shull, Smith, Uzzell, and Wang on the local bipartite covering numbers of difference graphs.In Chapter 4, we prove some bounds on the local dimension of any pair of layers of the Boolean lattice. In particular, we show that the local dimension of the first and middle layers is asymptotically n / log n.In Chapter 5, we introduce a new poset parameter called local t-dimension. We also discuss the fractional variants of this and other dimension-like parameters.All of Part I is joint work with Antnio Giro of the University of Cambridge and Kamil Popielarz of the University of Memphis.Chapter 2 of Part II is joint work with Victor Souza of IMPA (Instituto de Matemtica Pura e Aplicada, Rio de Janeiro).Chapter 3 of Part II is joint work with Antnio Giro
Turán-Type Results for Complete h-Partite Graphs in Comparability and Incomparability Graphs
This is the final version of the article. It was first available from Springer via http://dx.doi.org/10.1007/s11083-015-9384-6We consider an h-partite version of Dilworth's theorem with multiple partial orders. Let P be a fi nite 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 A₁, ..., Aₕ such that A₁ <ⱼ ... <ⱼ Aₕ holds for some 1 ≤ j ≤ r, and |A₁| = ... = |Aₕ| = Ω(|P|).
Also, we show that if the complement of G(P, n¹⁻ᵞ⁽α⁾ , where γ(α) → 0 as α → 1.
We provide a few applications of these results in combinatorial geometry, as well