Complete subgraphs of the coprime hypergraph of integers II: structural properties

Complete subgraphs of the coprime hypergraph of integers III: construction

“…Repeat the second step until it is not possible anymore. 4. Denote the resulting matching by M(n) and the set of not matched elements from P( 3 √ n, √ n) by F (n).…”

“…Similarly as in [4] we will consider a transformation which preserves the condition o p (A) ≤ k, thus, mapping shelves onto shelves. We will use a specific order ≺ on N >1 to make sure that this operation is idempotent (which facilitates the formulation).…”

“…In this third part of a series of three articles we continue the work in [3,4]. References and notation can be found there.…”

“…In [3] questions concerning the cardinality of maximum shelves were considered. The maximal number of prime divisors of elements in (maximum) shelves was investigated in [4]. It was shown that there exists a maximum shelf containing only elements with at most two prime divisors.…”

“…We should mention that the "direct" approach of expanding a maximum (n, k)-shelf does in general not lead to a maximum (m, k)-shelf for m > n. For example the maximum (6, 2)shelf {1, 2, 3, 5, 6} is no subset of any maximum (m, 2)-shelf for m ≥ 9, since the number 6 is not contained in any of those shelves. Using results from [4] the following can be shown (we denote by π the prime counting function, by ω(a) the number of prime divisors and by ssp(a) the second smallest prime divisor of a):…”