Untouchable Sets of Size 2q±1 in PG(2,q)

ABSTRACT An untouchable set in a projective plane is a set of points such that no line of the plane meets the set in exactly one point. Recently, Héger and Nagy (Avoiding Secants of Given Size in Finite Projective Planes, J. Combin. Des . 33:83–93, 2024.) provided a generalization of untouchable sets to ‐avoiding sets, and addressed the issue of the spectrum of sizes that such sets can attain in finite planes. In the case of untouchable sets, the authors state as an open question the existence of untouchable sets of size and . We answer this question in the affirmative for Desarguesian planes of even order, and provide a construction of untouchable sets of size in for .