The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is n while its local dimension is only 3.Hiraguchi (1955) proved that the maximum dimension of a poset of order n is n/2. However, we find a very different result for local dimension, proving a bound of Θ(n/ log n). This follows from connections with covering graphs using difference graphs which are bipartite graphs whose vertices in a single class have nested neighborhoods.We also prove that the local dimension of the n-dimensional Boolean lattice is Ω(n/ log n) and make progress toward resolving a version of the removable pair conjecture for local dimension.2010 Mathematics Subject Classification. 06A07,05C70.

Let $k>1$, and let $\mathcal{F}$ be a family of $2n+k-3$ non-empty sets of edges in a bipartite graph. If the union of every $k$ members of $\mathcal{F}$ contains a matching of size $n$, then there exists an $\mathcal{F}$-rainbow matching of size $n$. Replacing $2n+k-3$ by $2n+k-2$, the result is true also for $k=1$, and it can be proved (for all $k$) both topologically and by a relatively simple combinatorial argument. The main effort is in gaining the last $1$, which makes the result sharp.

Given a graph G, the non-cover complex of G is the combinatorial Alexander dual of the independence complex of G. Aharoni asked if the non-cover complex of a graphverified Aharoni's question in the affirmative for chordal graphs, we prove that the answer to the question is yes for all graphs. Namely, we show that for a graph G, the non-cover complex of a graph G is (|V (G)| − iγ(G) − 1)-collapsible.

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