Essentialities in additive bases

Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A��is no longer an asymptotic basis of any order and such that P is minimal among all subsets of A with this property. A finite essentiality of A is called an essential subset. In a recent paper, Deschamps and Farhi asked the following two questions : (i) does every asymptotic basis of N_0 possess some essentiality ? (ii) is the number of essential subsets of size at most k of an asymptotic basis of order h bounded by a function of k and h only (they showed the number is always finite) ? We answer the latter question in the affirmative, and the former in the negative by means of an explicit construction, for every integer h >= 2, of an asymptotic basis of order h with no essentialities.
Version 4 : 5 pages. Theorem 2.2 is new. The title of the paper and the abstract have been changed to reflect more accurately the additional content. This version has been submitted for publication, so no further changes are anticipated