15 problems

Let $q$ be a number, and let $G$ be an Abelian group with … A subset $B\subset G$ is a difference set if every non-unit element of $G$ has a unique representation $g=ab^{-1}$ with…

Let $q$ be even, let $k$ be neither too small nor too large, and let $m$ be a possible cardinality. A point set is $k$-avoiding modulo $2$ when every secant length is different fro…

Let $\mathrm{PG}(2,q)$ be the Desarguesian projective plane of order $q$, and let $\mathrm{Sp}(k,q)$ be the set of cardinalities of point sets avoiding $k$-secants. Complete-spectr…

Let $q$ be an even integer, and let $\mathrm{PG}(2,q)$ denote the Desarguesian projective plane of order $q$. A $(3q+4,4)$-arc is a point set of size $3q+4$ meeting every line in a…

Let $\Pi_q$ be a projective plane of order $q$, and let $\mathrm{Sp}(k,q)$ denote the spectrum of cardinalities of point sets avoiding $k$-secants. The interval…

Kriesell's conjecture. The projective plane $\mathbb{P}^2(\mathbb{F}_q)$ can be partitioned into

Let $X$ be a simplicial complex on $n$ vertices, and suppose that … Suppose also that $X$ admits a free $\mathbb{Z}_{n}$-action. Vertex-minimality conjecture. The complex $X_q$ is…

Spectral gap conjecture. For $1\leq k\leq\sqrt{p-2}$, the intervals

Let $P_n=(P,L)$ be a finite projective plane of order $n$, and let $chi(P_n)$ denote the minimum number of colors in a legitimate coloring, meaning a coloring of the points for whi…

Let $\Pi_q$ be a finite projective plane of order $q$, let $T_r(\Pi_q)$ denote the maximum percolation time among percolating sets at infection rate $r$, and let $r_1$ and $r_2$ be…

Let $\Pi_q$ be a finite projective plane of order $q$, let $r$ be the infection rate, and let $t_r(A)$ denote the percolation time of a percolating set $A$. Two-round minimal perco…

Let $\Pi_q$ be a finite projective plane of order $q$, let $r$ be the infection rate, and let $M_r(\Pi_q)$ denote the maximum cardinality of a nonpercolating set of points. Maximum…

Let $(P,L)$ be a linear space with $d=\#P$, and let $A$ be its associated ring. Set $n$ and $m$ to be the minimum and maximum, respectively, size of a line in $L$. Suppose…

Conjecture. Equality holds:

Let $q$ be the order of the finite projective plane, and let $f(n)$ denote the minimum number of lines whose odd-intersection point set has size $n$. Conjecture. … The paper has es…