Local dimension is unbounded for planar posets

In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height

Boolean Dimension, Components and Blocks

We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if $\dim(C)\le d$ for every component $C$ of a poset $P$, then $\dim(P)\le \max\{2,d\}$; also if $\dim(B)\le d$ for every block $B$ of a poset $P$, then $\dim(P)\le d+2$. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if $\text{ldim}(C)\le d$ for every component $C$ of a poset $P$, then $\text{ldim}(P)\le d+2$; however, for every $d\ge 4$, there exists a poset $P$ with $\text{ldim}(P)=d$ and $\dim(B)\le 3$ for every block $B$ of $P$. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if $\text{bdim}(C)\le d$ for every component $C$ of $P$, then $\text{bdim}(P)\le 2+d+4\cdot2^d$; also if $\text{bdim}(B)\le d$ for every block of $P$, then $\text{bdim}(P)\le 19+d+18\cdot 2^d$.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1712.0609

Power Laws Variance Scaling of Boolean Random Varieties

International audienceLong fibers or stratifed media show very long range correlations. These media can be simulated by models of Boolean random varieties and their iteration. They show non standard scaling laws with respect to the volume of domains K for the variance of the local volume fraction: on a large scale, the variance of the local volume fraction decreases according to power laws of the volume of K. The exponent is equal to (n-k)/ n for Boolean varieties with dimension k in the space Rn : 2/3 for Boolean fibers in 3D, and 1/3 for Boolean strata in 3D. When working in 2D, the scaling exponent of Boolean fibers is equal to 1/2. From the results of numerical simulations, these scaling laws are expected to hold for the prediction of the effective properties of such random media

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

Bounding Embeddings of VC Classes into Maximum Classes

One of the earliest conjectures in computational learning theory-the Sample Compression conjecture-asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes---those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving the conjecture is by embedding general VC classes into maximum classes without super-linear increase to their VC dimensions, as such embeddings would extend the known compression schemes to all VC classes. We show that maximum classes can be characterised by a local-connectivity property of the graph obtained by viewing the class as a cubical complex. This geometric characterisation of maximum VC classes is applied to prove a negative embedding result which demonstrates VC-d classes that cannot be embedded in any maximum class of VC dimension lower than 2d. On the other hand, we show that every VC-d class C embeds in a VC-(d+D) maximum class where D is the deficiency of C, i.e., the difference between the cardinalities of a maximum VC-d class and of C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible results on embedding into maximum classes. For some special classes of Boolean functions, relationships with maximum classes are investigated. Finally we give a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum classes for smallest k.Comment: 22 pages, 2 figure