12 problems

Fisher's conjecture. The average size of a complete arc is about

Kim–Vu conjecture. The constant can be reduced to $C=10$, so that

Let $t_{2}(2,q)$ be the smallest size of a complete arc in $\mathrm{PG}(2,q)$, and let the bounds denoted in the source by … be the explicit upper bounds obtained from the computed…

Probabilistic upper-bound conjecture. These upper bounds hold for all $q$ without any extra conditions. The estimates originate from a conjecture about the unproved steps of a gree…

Let $\mathcal{P}_{q}$ be a projective plane of order $q$, and let $t(\mathcal{P}_{q})$ denote the size of its smallest complete arc. For sufficiently large $q$, suppose that … wher…

Greedy-algorithm bounds conjecture. These upper bounds hold for all $q$ without any extra conditions.

Let $t_{2}(2,q)$ denote the smallest size of a complete arc in the projective plane $\mathrm{PG}(2,q)$. The bounds … … and … as well as … should hold for all $q\geq109$. These are…

Spectrum conjecture. For every prime power $q$ with $353\leq q\leq4561$ and $q\in T_2\cup T_3$, $PG(2,q)$ has complete $k$-arcs of all sizes

Let $\overline{H}_{q}$, $\overline{G}_{q}$, and $\overline{J}_{q}$ be the quantities defined in the paper. Assume that $q$ is prime for $\overline{H}_{q}$, a prime power for…

Let $PG(2,q)$ be the projective plane of order $q$. Constructions A, B, and C produce families of complete $k$-arcs in the size regions stated in the paper's theorem: Construction…

Let $PG(2,q)$ be the projective plane of order $q$, and let $t_{2}(2,q)$ denote the smallest size of a complete arc in $PG(2,q)$. Complete-arc bounds conjecture. The estimates … an…

Let $PG(2,q)$ be the projective plane of order $q$, and let $t_{2}(2,q)$ denote the smallest size of a complete arc in $PG(2,q)$. Logarithmic upper-bound conjecture. For every…