Untouchable sets of size $2q \pm 1$ 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, H\'eger 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 $k$-avoiding sets, and addressed the issue of the spectrum of sizes that such sets can attain in finite planes. Specific to the untouchable set case, the authors state as an open question the existence of untouchable sets of size $2q-1$ and $2q+1$. We answer this question in the affirmative for Desarguesian planes of even order, and provide a construction of untouchable sets of size $2q+1$ in $PG(2,q)$ for $q \equiv 3\pmod{4}$.