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Let T(r; n) denote the maximum number of subsets of an n-set satisfying the condition in the title. It is proved in a purely combinatorial way, that for n suuciently large log 2 T(r; n) n 8 log 2 r r 2 holds.

Improving a result of Erd˝ os, Gyárfás and Pyber for large n we show that for every integer r 2 there exists a constant n 0 = n 0 (r) such that if n n 0 and the edges of the complete graph K n are colored with r colors then the vertex set of K n can be partitioned into at most 100r log r vertex disjoint monochromatic cycles.

Let f d (G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n > n 0 (D) vertices, f 2 (G) = n − D − 1 and f 3 (G) ≥ n − O(D 3). For d ≥ 4, f d (G) depends strongly on the actual structure of G, not only on the maximum degree of… (More)

We show in this paper that in every 3-coloring of the edges of K n all but o(n) of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochromatic… (More)

A family of n-dimensional unit norm vectors is a Euclidean superimposed code, if the sums of any two distinct at most m-tuples of vectors are separated by a certain minimum Euclidean distance d. Ericson and Györfi [8] proved that the rate of such a code is between (log m)/4m and (log m)/m for m large enough. In this paper – improving the above longstanding… (More)

Let c be a positive constant. We show that if r = cn 1/3 and the members of [n] r are chosen sequentially at random to form an intersecting hypergraph then with limiting probability (1 + c 3) −1 , as n → ∞, the resulting family will be of maximum size n−1 r−1 .