Levon H. Khachatrian

The Complete Intersection Theorem for Systems of Finite Sets

We are concerned here with one of the oldest problems in combinatorial extremal theory. It is readily described after we have made a few conventions. ‫ގ‬ denotes the set of

On Perfect Codes and Related Concepts

The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codesattaining the sphere–packing bound) is introduced. This was motivated by the “code–anticode” bound of Delsartein distance regular graphs. This bound in conjunction with the recent complete solutions of diametric problems in the Hamming graph ℌq(n) and the Johnson graph J(n,k)gives a sharpening of the sphere–packing bound. Some necessaryconditions for the existence of diameter perfect codes are given.In the Hamming graph all diameter perfect codes over alphabetsof prime power size are characterized. The problem of tilingof the vertex set of J(n,k) with caps (and maximalanticodes) is also examined.

The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets

The authors have proved in a recent paper a complete intersection theorem for systems of finite sets. Now we establish such a result for nontrivial-intersection systems (in the sense of Hilton and Milner Quart. J. Math. Oxford18(1967), 369?384].

A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated.

Fault‐tolerant minimum broadcast networks

Broadcasting is the task of transmitting a message originated at one processor of a communication network to all other processors in the network. A minimal k-fault-tolerant broadcast network is a communication network on n vertices in which any processor can broadcast in spite of up to k line failures in optimal time T-n(k). In this paper, we study B-k(n), the minimum number of communication lines of any minimal k-fault-tolerant broadcast network on n processors. We give the value of B-k(n) for several values of n and k and, in case k < [log n], give almost-minimum k-fault-tolerant broadcast networks

Maximum Number of Constant Weight Vertices of the Unit n -Cube Contained in a k -Dimensional Subspace

We introduce and solve a natural geometrical extremal problem. For the set E (n,w) = {xn ∈ {0,1}n : xn has w ones } of vertices of weight w in the unit cube of ℝn we determine M (n,k,w) ≜ max{|Ukn ∩ E(n,w)|:Ukn is a k-dimensional subspace of ℝn . We also present an extension to multi-sets and explain a connection to a higher dimensional Erdős–Moser type problem.

Messy broadcasting in networks

In the classical broadcast model it is tacitly assumed that every member of the scheme produces the broadcasting in the most clever way, assuming either that there is a leader or a coordinated set of protocols. In this paper, we assume that there is no leader and that the state of the whole scheme is secret from the members; the members do not know the starting time and the originator and their protocols are not coordinated. We consider three new models of broadcasting, which we call “Messy broadcasting.”

Katona’s Intersection Theorem: Four Proofs

It is known from a previous paper [3] that Katona’s Intersection Theorem follows from the Complete Intersection Theorem by Ahlswede and Khachatrian via a Comparison Lemma. It also has been proved directly in [3] by the pushing–pulling method of that paper. Here we add a third proof via a new (k,k+1)-shifting technique, whose impact will be exploared elsewhere. The fourth and last of our proofs is a gift from heaven for Gyula’s birthday.

Extremal problems under dimension constraints

The aim of this paper is to systematically present an area of extremal problems under dimension constraints. We state conjectures and solutions for many of these problems. Proofs will be given in several papers, each devoted to a specific problem.

Counterexample to the Frankl-Pach conjecture for uniform, dense families

N denotes the set of positive integers and for e, nEN, e<n we set 2[n]={F:FC[1,n]}, ([~]) = {FE2[n]:IF[:g}. family 5~ C 2 [n] is called g-dense, e C N, if there exists an g-element subset D E ([~]) A with .~(D)={FAD:FE~} satisfying (1) ]~(D)[ = 2 e. A well-known result of Sauer [1], Shelah-Perles [2], and Vapnik-Cervonenkis [a] says that any 5 ~ C 2 In] is e-dense, if i<g Frankl-Pach [4] proved that any e-uniform ~, that is ~C ([~]), is e-dense, if