Critical numbers of natural intervals

In this paper we study the critical numbers cr(r,n)cr(r,n) of natural intervals [r,n][r,n]. The critical number cr(1,n)cr(1,n) is the smallest integer l   satisfying the following conditions: (i) every sequence of integers S={r1=1⩽r2⩽⋯⩽rk}S={r1=1⩽r2⩽⋯⩽rk} with r1+⋯+rk=nr1+⋯+rk=n and k⩾lk⩾l has the following property: every integer between 1 and n can be written as a sum of distinct elements of S, and (ii) there exists S   with k=lk=l, satisfying this property. The definition of cr(r,n)cr(r,n) for r>1r>1 is a natural extension of cr(1,n)cr(1,n). We completely determined the values of cr(1,n)cr(1,n) and cr(2,n)cr(2,n). For r>2r>2, we determined the values of cr(r,n)cr(r,n) for n>3r2n>3r2. Similar problems concerning subsets of finite groups were introduced by Erdös and Heilbronn in 1964 and extended by other authors.