Untouchable sets of size $2q \pm 1$ in $PG(2,q)$
Jeremy M. Dover
TL;DR
The paper addresses the existence of untouchable sets of sizes $2q-1$ and $2q+1$ in $PG(2,q)$. It uses explicit conic-pencil constructions in Desarguesian planes to build untouchable sets, with separate treatments for even $q$ and odd $q$. For even $q$, it produces a $2q-1$ set via $C_a\cup C_{a^2}\cup {(1,1,a^2)}$ and a $2q+1$ set via $D_a\cup D_b\cup {(0,1,a),(0,1,b)}$; a $2q-2$ case also appears when $a^4=a$. For odd $q$, a $2q+1$ set is obtained using mutually exterior conics $C_1$ and $C_b$ when $q \equiv 3 \pmod{4}$, with a counting argument proving untouchability; the $q \equiv 1 \pmod{4}$ case remains unresolved. Overall, the results corroborate the conjectured presence of sizes $2q \pm 1$ in the spectrum for many $q$, while highlighting remaining gaps and methodological limits of conic pencils.
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 $k$-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 $q \equiv 3\pmod{4}$.