Let f(m,n) denote the number of relatively prime subsets of {m + 1,m + 2, . . . , n}, and let Φ(m,n) denote the number of subsets A of {m+1,m+2, . . . , n} such that gcd(A) is relatively prime to n. Let fk(m,n) and Φk(m,n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A nonempty s...

Let n be a positive integer and let A be a nonempty finite set of positive integers. We say that A is relatively prime if gcd(A) = 1, and that A is relatively prime to n if gcd(A, n) = 1. In this work we count the number of nonempty subsets of A that are relatively prime and the number of nonempty subsets of A that are relatively prime to n. Related formulas are also obtained for the number of ...

A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A are each nonempty, and A ∩ B ∩ C = ∅. Let A be a family of r-element subsets of an n-element set, containing no triangle. Our main result implies that for r ≥ 3 and n ≥ 3r/2, we have |A| ≤ ( n−1 r−1 ) . This settles a longstanding conjecture of Erdős [7], by improving on earlier results of Bermond, Chvátal, Frankl, and Füredi. W...

the neighbourhood polynomial g , is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph. in other word $n(g,x)=sum_{uin n(g)} x^{|u|}$, where n(g) is neighbourhood complex of a graph, whose vertices are the vertices of the graph and faces are subsets of vertices that have a common neighbour. in this paper we compute this polynomial for some na...

Abstract We prove the existence of an optimal domain for minimizing buckling load among all, possibly unbounded, open subsets $${\mathbb {R}}^n$$ R n ( $$n\ge 2$$ ≥ 2 ) with given measure. Our approach is based on...

we denote by $ls[n](t,k,v)$ a large set of $t$-$(v,k,lambda)$ designs of size $n$, which is a partition of all  $k$-subsets ofa $v$-set into $n$ disjoint  $t$-$(v,k,lambda)$ designs and$n={v-t choose k-t}/lambda$.   we use the notation$n(t,v,k,lambda)$ as the maximum possible number of mutuallydisjoint cyclic $t$-$(v,k,lambda)$designs. in this paper we givesome new bounds for $n(2,29,4,3)$ and ...