A scattering of orders

A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class $\mathcal B$ of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in $\mathcal B$. More generally, we say that a partial ordering is $\kappa$-scattered if it does not contain a copy of any $\kappa$-dense linear ordering. We prove analogues of Hausdorff's result for $\kappa$-scattered linear orderings, and for $\kappa$-scattered partial orderings satisfying the finite antichain condition. We also study the $\mathbb{Q}_\kappa$-scattered partial orderings, where $\mathbb{Q}_\kappa$ is the saturated linear ordering of cardinality $\kappa$, and a partial ordering is $\mathbb{Q}_\kappa$-scattered when it embeds no copy of $\mathbb{Q}_\kappa$. We classify the $\mathbb{Q}_\kappa$-scattered partial orderings with the finite antichain condition relative to the $\mathbb{Q}_\kappa$-scattered linear orderings. We show that in general the property of being a $\mathbb{Q}_\kappa$-scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions

Spectral gap for random-to-random shuffling on linear extensions

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling.Comment: 16 pages, 10 figures; v2: typos fixed plus extra information in figures; v3: added explicit conjecture 2.2 + Section 3.6 on the diameter of the Markov Chain as evidence + misc minor improvements; v4: fixed bibliograph

A Family of Partially Ordered Sets with Small Balance Constant

Given a finite poset $\mathcal P$ and two distinct elements $x$ and $y$, we let $\operatorname{pr}_{\mathcal P}(x \prec y)$ denote the fraction of linear extensions of $\mathcal P$ in which $x$ precedes $y$. The balance constant $\delta(\mathcal P)$ of $\mathcal P$ is then defined by $$\delta(\mathcal P) = \max_{x \neq y \in \mathcal P} \min \left\{ \operatorname{pr}_{\mathcal P}(x \prec y), \operatorname{pr}_{\mathcal P}(y \prec x) \right\}.$$ The $1/3$-$2/3$ conjecture asserts that $\delta(\mathcal P) \ge \frac13$ whenever $\mathcal P$ is not a chain, but except from certain trivial examples it is not known when equality occurs, or even if balance constants can approach $1/3$. In this paper we make some progress on the conjecture by exhibiting a sequence of posets with balance constants approaching $\frac{1}{32}(93-\sqrt{6697}) \approx 0.3488999$, answering a question of Brightwell. These provide smaller balance constants than any other known nontrivial family.Comment: 11 pages, 4 figure

Length of an intersection

A poset \bfp is well-partially ordered (WPO) if all its linear extensions are well orders~; the supremum of ordered types of these linear extensions is the {\em length}, \ell(\bfp) of \bfp. We prove that if the vertex set $X$ of \bfp is infinite, of cardinality $\kappa$, and the ordering $\leq$ is the intersection of finitely many partial orderings $\leq_i$ on $X$, $1\leq i\leq n$, then, letting \ell(X,\leq_i)=\kappa\multordby q_i+r_i, with $r_i<\kappa$, denote the euclidian division by $\kappa$ (seen as an initial ordinal) of the length of the corresponding poset~: \ell(\bfp)< \kappa\multordby\bigotimes_{1\leq i\leq n}q_i+ \Big|\sum_{1\leq i\leq n} r_i\Big|^+ where $|\sum r_i|^+$ denotes the least initial ordinal greater than the ordinal $\sum r_i$. This inequality is optimal (for $n\geq 2$).Comment: 13 page

Chain Decomposition Theorems for Ordered Sets (and Other Musings)

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a ``strong'' way, is proved. The result is motivated by a conjecture of Graham's concerning probability correlation inequalities for linear extensions of finite posets