On a problem of Ahlswede and Katona

Abstract

Abstract. Let p(G) denote the number of pairs of adjacent edges in a graph G. Ahlswede and Katona considered the problem of maximizing p(G) over all simple graphs with a given number n of vertices and a given number N of edges. They showed that p(G) is either maximized by a quasi-complete graph or by a quasi-star. They also studied the range of N (depending on n) for which the quasi-complete graph is superior to the quasi-star (and vice versa) and formulated two questions on distributions in this context. This paper is devoted to the solution of these problems.

Probably Intersecting Families are Not Nested

“…Moreover, they showed that there exists some non-negative integer R (depending on n) such that for i < 1 2 n 2 − R and for 1 2 n 2 i 1 2 n 2 + R the extremal family is a quasi-star, while for all other values of i the extremal family is a quasi-complete graph. Wagner and Wang [6] extended this by finding the value of R explicitly and showing that it is non-zero for a proportion √ 2 − 1 of numbers n. Combining Theorem 1.3 with these results we see that the extremal families even for N in this range are surprisingly complicated: for many values of n (i.e., those for which R = 0) the extremal families can switch between the two classes three times just in this single layer.…”