An isoperimetric inequality for antipodal subsets of the discrete cube

A family of subsets of $\{1,2,\ldots,n\}$ is said to be {\em antipodal} if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of $\{1,2,\ldots,n\}$. Our inequality implies that for any $k \in \mathbb{N}$, among all such families of size $2^k$, a family consisting of the union of a $(k-1)$-dimensional subcube and its antipode has the smallest possible edge boundary.Comment: A new proof of Lemma 6 (kindly suggested by an anonymous referee) has been given; this shortens our original argument. An acknowledgement and a conclusion have been added, and minor changes have been made to improve readabilit

A collection of open problems in celebration of Imre Leader's 60th birthday

One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics

Isoperimetry and volume preserving stability in real projective spaces

We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset \mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritor\'{e} and A. Ros on $\mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $\mathbb{S}^n$.Comment: 19 pages, 1 figur

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed

The cone volume measure of antipodal points

The optimal condition of the cone volume measure of a pair of antipodal points is proved and analyzed. © 2015, Akadémiai Kiadó, Budapest, Hungary

Hyperbolicity, Assouad-Nagata dimension and orders on metric spaces

We study asymptotic invariants of metric spaces, defined in terms of the travelling salesman problem, and our goal is to classify groups and spaces depending on how well they can be ordered in this context. We characterize virtually free groups as those admitting an order which has some efficiency on $4$-point subsets. We show that all $\delta$-hyperbolic spaces can be ordered extremely efficiently, for the question when the number of points of a subset tends to $\infty$. We prove that all spaces of finite Assouad-Nagata dimension admit a good order for the above mentioned problem, and under an additional hypothesis we prove the converse. Despite travelling salesman terminology, our paper does not aim at applications in computer science. Our goal is to study new properties of groups and metric spaces, and describe their connection with more traditional invariants, such as hyperbolicity, dimension, number of ends and doubling.Comment: title and abstract revised. minor correction

Erd}os-Ko-Rado type problems, discrete isoperimetric inequalities and other problems in extremal combinatori.

PhD ThesisIn this thesis, we investigate three problems in extremal combinatorics, using methods from combinatorics, representation theory, nite eld theory and probabilistic combinatorics. Firstly, for a prime power q and a positive integer n, we say a subspace U of Fnq is cyclically covering if the union of the cyclic shifts of U is equal to Fnq . We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of Fnq . This is a natural generalisation of a problem posed in 1991 by Cameron. We prove several upper and lower bounds, and for each xed q, we answer the question completely for in nitely many values of n (which take the form of certain geometric series). Our results imply lower bounds for a well-known conjecture of Isbell, and a generalisation thereof, supplementing lower bounds due to Spiga. We also consider the analogous problem for general representations of groups, and also provide some results for natural representations of the symmetric group Sn. Second, for positive integers n and r, we let Qr n denote the rth power of the n-dimensional discrete hypercube graph (i.e., vertex set f0; 1gn and edges between 0-1 vectors separated by Hamming distance at most r). We study edge isoperimetric inequalities for this graph. Harper, Bernstein, Lindsey and Hart proved a best-possible edge isoperimetric inequality for this graph in the case r = 1. For each r 2, we obtain an edge isoperimetric inequality for Qr n; our inequality is tight up to a constant factor depending only upon r. Our techniques also give an edge isoperimetric inequality for the `Kleitman-West graph' (the graph whose vertices are the k-element subsets of f1; 2; :::; ng, where two k-element sets are joined by an edge if they have symmetric di erence of size 2); this inequality is tight up to a factor of 2 + o(1) for sets of size ns ks , where k = o(n) and s 2 N. Finally, for positive integers n and d, we say sets A;B [n] are d-close if the minimum 5 cyclic distance between some element a 2 A and some element b 2 B is at most d. We say a set system A P([n]) is d-close if every pair of sets A;B 2 A is d-close, and a say a pair of set systems A; B is cross d-close if every pair of sets A 2 A;B 2 B is d-close. We investigate the maximum possible sizes of such set systems, particularly for each non-negative integer k, the maximum possible size of k-uniform d-close set systems (i.e., d-close set systems A [n](k)). This is a natural extension of the well-known result of Erd}os, Ko and Rado, which corresponds to the case d = 0. We prove tight, stable bounds for k at most a constant fraction of n depending on d. To do so we employ the junta method, introduced to extremal combinatorics by Dinur and Friedgut, and initially applied to Erd}os-Ko-Rado type problems by Keller and Lifshitz