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A classical topic in combinatorics is the study of problems of the following type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections F n F of any F and F' in F are: (i) FnF'# 0, where all FEF have k(More)
The covering radius R of a code is the maximal distance of any vector from the code. This work gives a number of new results concerning t[ n, k], the minimal covering radius of any binary code of length n and dimension k. For example r[ n, 41 and t [ n, 51 are determined exactly, and reasonably tight bounds on t[ n, k] are obtained for any k when n is(More)
The present paper represents essentially a chapter in a forthcoming "Monographie" in the l Enseignement Mathématique series i) with the title "Old and new problems and results in combinatorial number theory" by the above authors. Basically we will discuss various problems in elementary number theory, most of which have a combinatorial flavor. In general we(More)
Let G be a finite connected graph. If x and y are vertices of G, one may define a distance function d, on G by letting d&x, y) be the minimal length of any path between x and y in G (with d&, x) = 0). Thus, for example, d&x, y) = 1 if and only if {x, y} is an edge of G. Furthermore, we define the distance matrix D(G) for G to be the square matrix with rows(More)
It is standard practice among authors discussing the minimum spanning tree problem to refer to the work of Kruskal(1956) and Prim (1957) as the sources of the problem and its first efficient solutions, despite the citation by both of Boruvka (1926) as a predecessor. In fact, there are several apparently independent sources and algorithmic solutions of the(More)
For a ®xed graph H, the Ramsey number r (H) is de®ned to be the least integer N such that in any 2-coloring of the edges of the complete graph K N , some monochromatic copy of H is always formed. Let r(n, Á) denote the class of graphs H having n vertices and maximum degree at most Á. It was shown by Chvata  l, Ro È dl, Szemere  di, and Trotter that for(More)