Generalized Hyperfocused Arcs in

Abstract

A generalized hyperfocused arc scriptH in PG(2,q) is an arc of size k with the property that the k(k−1)/2 secants can be blocked by a set of k−1 points not belonging to the arc. We show that if q is a prime and scriptH is a generalized hyperfocused arc of size k, then k=1,2, or 4. Interestingly, this problem is also related to the (strong) cylinder conjecture ( Problem 919), as we point out in the last section.

A One-Page Solution of a Problem of Erdős and Purdy

“…An algebraic proof of Conjecture 1 is given in [10]. Conjecture 1 was considered also over F p and there it was proved by Blokhuis et al [2].…”