Dmitry Fon-Der-Flaass

Orbits of antichains revisited

Abstract We present here a new treatment of the permutation f of antichains in ranked posets moving the set of lower units of a monotone Boolean function to the set of its upper zeros. Shorter and more transparent proofs for some known properties of f are presented. The orbits of f for a direct product of three chains are considered in some detail.

Orbits of Antichains in Ranked Posets

We consider the permutation f of antichains of a ranked poset P , moving the set of lower units of any monotone boolean function on P to the set of its upper zeros. A duality relation on orbits of this permutation is found, which is used for proving a conjecture by M. Deza and K. Fukuda. For P a direct product of two chains, possible lengths of orbits are completely determined.

Covering boxes by points

Abstract Families of boxes in the d -dimensional Euclidean space are considered. A simple proof of an upper bound on the size of a minimum transversal in terms of the space dimension and the independence number of the intersection graph of the family is given. The proof uses an idea by Gyarfas and Lehel. Some examples are constructed, showing that in three instances this upper bound is exact.

Bases for permutation groups and matroids

Abstract In this paper, we give two equivalent conditions for the irredundant bases of a permutation group to be the bases of a matroid. (These are deduced from a more general result for families of sets.) If they hold, then the group acts geometrically on the matroid, in the sense that the fixed points of any element form a flat. Some partial results towards a classification of such permutation groups are given. Further, if G acts geometrically on a perfect matroid design, there is a formula for the number of G -orbits on bases in terms of the cardinalities of flats and the numbers of G -orbits on tuples. This reduces, in a particular case, to the inversion formula for Stirling numbers.

On Deeply Critical Oriented Graphs

For every positive integer k, we present an oriented graph Gk such that deleting any vertex of Gk decreases its oriented chromatic number by at least k and deleting any arc decreases the oriented chromatic number of Gk by two.

There Exists no Distance-regular Graph with Intersection Array (5,4,3; 1,1,2)

The problem of deciding whether a distance-regular graph with a given set of parameters exists in a very hard one, and far from solution. The set of parameters mentioned in the title is the smallest unsolved case 1. We will give here a short, computer-free proof that there is no graph with these parameters.

A Note on Greedy Codes

In their paper (J. Combin. Theory Ser. A64(1993), 10?30) Brualdi and Pless prove linearity of some binary codes obtained by a greedy algorithm and establish lower bounds for the dimension of these codes. In this note, we show that actually they have proved a much more general result, and show that these codes also satisfy the Varshamov?Gilbert bound.

Cartesian Products of Graphs and Metric Spaces

We prove uniqueness of decomposition of a finite metric space into a product of metric spaces for a wide class of product operations. In particular, this gives the positive answer to the long-standing question of S. Ulam: `IfU×U?V×V with U,V compact metric spaces, will then U and V be isometric?? in the case of finite metric spaces. In the proof we use uniqueness of cartesian decomposition of connected graphs; a known fact to which we give a new proof which is shorter and more transparent than existing ones.

A combinatorial construction for twin trees

Abstract The notions of a building [6] and, later, a twin building [7] were introduced to give an axiomatical geometric treatment of groups with BN-pairs and with twin BN-pairs. For this reason, buildings and twin buildings have always been studied from a point of view relevant to (and originated from) BN-pairs. The notion of a twin tree, an affine twin building of rank 2, was introduced in [2] by a self-contained, very elementary and geometric definition. The study of twin trees in [2–4], and the explicit constructions given there, also use notions and groups inspired by BN-pairs. In this paper we shall present an elementary combinatorial construction for twin trees which, in a sense, is orthogonal to the BN-pairs approach. This construction gives all twin trees; in particular, it provides an easy proof that twin trees of any given valency are uncountably many (a result proved independently by M. Ronan [1]). Our approach turns out to be especially fruitful for twin trees of valency 3 (and probably of valency 4; that part of the work is still in progress). In particular, it highlights a new geometric object ( a pairing) within a twin tree—and hence a new class of subgroups of its automorphism group. When the twin tree in question has a ‘rich’ automorphism group, study of these objects (and corresponding subgroups) gives a lot of new and unexpected information on combinatorial and group-theoretic structure of the twin tree.

On Exponential Trees

We give a necessary and sufficient condition for a tree from a certain class to have exponential growth rate (in the sense of 1). The class contains, in particular, all trees of bounded valency; and also includes the class of trees without end-vertices which was considered in 1.