9 problems

Let ${\textup{PG}}(2,q^3)$ be a projective plane with $q$ even and $q>2$. Let $\mathcal{C}$ be a conic with nucleus $N$ tangent to the line $\ell_\infty$ at $T$, and let…

Multi-arc smoothing lower-bound conjecture. The multi-arc $calpha$ has a smoothing that cannot be homotoped to a multi-arc with fewer than

Let $2\le N\le q-2$. In $\mathrm{PG}(N,q)$, a normal rational curve is a $(q+1)$-arc projectively equivalent to … A $(q+1)$-arc is complete if no further point can be added while p…

The $\sqrt{q}$ conjecture. The second largest complete arc in ${\mathrm{PG}}(2,q)$ has size

Weight-one vector conjecture. The matrix $\mathrm{M}_{n}$ has a vector of weight one in its column space.

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 $PG(2,q)$. The finite-range upper-bound conjecture. In…

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 $PG(2,q)$. The logarithmic upper-bound conjecture. In…

Let $PG(2,q)$ be the projective plane of order $q$, let $\overline{t}_{2}(2,q)$ be the smallest known size of a complete arc, and let $M_q$ be the upper endpoint defined in the sou…

Let $q$ be even, and let a complete arc in the projective plane $PG(2,q)$ mean an arc that cannot be enlarged by adding another point. Two arcs are projectively equivalent when a p…