On separating systems whose elements are sets of at most k elements

Completely separating systems of k-sets

Dickson On a problem concerning separating systems of a nite set, Journal of Combinatorial Theory, 7 (1969), 191{196.] introduced the notion of a completely separating set system. We study such systems with the additional constraint that each set in the system has the same size. Let T denote an n-set. We say that a subset S of T separates i from j if i 2 S and j 6 2 S. A collection of k-sets C ...

Completely separating systems of k-sets for 7 ≤ k ≤ 10

Spanning trees whose stems have at most k leaves

Let T be a tree. A vertex of T with degree one is called a leaf, and the set of leaves of T is denoted by Leaf(T ). The subtree T − Leaf(T ) of T is called the stem of T and denoted by Stem(T ). A spanning tree with specified stem was first considered in [3]. A tree whose maximum degree at most k is called a k-tree. Similarly, a stem whose maximum degree at most k in it is called a k-stem, and ...

On the size of sets whose elements have perfect power n - shifted products

We show that the size of sets A having the property that with some non-zero integer n, a1a2 + n is a perfect power for any distinct a1, a2 ∈ A, cannot be bounded by an absolute constant. We give a much more precise statement as well, showing that such a set A can be relatively large. We further prove that under the abcconjecture a bound for the size of A depending on n can already be given. Ext...

Which elements of a finite group are non-vanishing?

‎Let $G$ be a finite group‎. ‎An element $gin G$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $G$‎, ‎$chi(g)neq 0$‎. ‎The bi-Cayley graph ${rm BCay}(G,T)$ of $G$ with respect to a subset $Tsubseteq G$‎, ‎is an undirected graph with‎ ‎vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin G‎, ‎ tin T}$‎. ‎Let ${rm nv}(G)$ be the set‎ ‎of all non-vanishi...