On multiple nuclei and a conjecture of Lunelli and Sce A. Blokhuis Dedicated to J. A. Thas on his ﬁftieth birthday Abstract We obtain new lower bounds on the size of a t-fold blocking set in AG(2,q), in the case that (t, q) = 1. As a consequence, we get that the Lunelli-Sce conjecture on the maximal size of a (k, n)-arc is true in the aﬃne plane. 1 Introduction Let A = AG(2,q ) be the desarguesian aﬃne plane of order q .A nucleus of a set S of q + 1 points of A, is a point P ∈ S , with the property, that every line through P meets the set S (exactly once). The main result [4] is, that a (q + 1)-set has at most q − 1 nuclei. The only known examples of sets having this number of nuclei, are a set consisting of a line together with a point outside, and a sporadic example in the plane of order 5, where the 10 points of a Desargues conﬁguration can be partitioned into sets of size 6 and 4, where the second set consists precisely of the nuclei of the ﬁrst. It appears to be a diﬃcult problem, to characterize the sets S with exactly this number of nuclei. Partial results in this direction were obtained in [3]. In [2], the notion of nucleus was extended to arbitrary sets. Here P is a (gen- eralized) nucleus of S , if P ∈ S , and every line through P contains a point of S . The main result was, that a set of size q + k has at most k(q − 1) nuclei, and again this result is best possible (for k<q ). As a corollary of this result, one obtains the Received by the editors in February 1994 AMS Mathematics Subject Classiﬁcation: Primary 51E21, Secondary 05B25 Keywords: Nuclei, t-fold blocking set, (k, n)-arcs. Bull. Belg. Math. Soc. 3 (1994), 349–353