9 problems

Let $C\!D(k,q)$ be a connected component of $D(k,q)$, and let $\lambda_2$ denote its second largest eigenvalue. Ustimenko's conjecture. For all $k$ and $q$, … This is a universal s…

Let $A(n,q)$ be the algebraically defined graph introduced in the paper, and let its second largest eigenvalue mean the second largest eigenvalue of its adjacency matrix. Nearly Ra…

Let $C\!D(k,q)$ be a connected component of $D(k,q)$. Schneider's conjecture. For every prime power $q$, $\operatorname{diam}(C\!D(3,q))=6$. For $k>3$ odd, the diameter is $k+5$, a…

Let $C\!D(k,q)$ be a connected component of $D(k,q)$, with $k\ge 2$ and $q$ a prime power. Lazebnik–Ustimenko–Woldar's conjecture. There is a positive constant $C$ such that … This…

Let $D(k,q)$ be the graph defined by the parameters $k$ and the prime power $q$. Lazebnik–Ustimenko–Woldar's conjecture. For every prime power $q\ge 4$, the graph has girth … The s…

Let $H$ be an irreducible affine hypersurface in $\mathbb{A}^s\times\mathbb{A}^s$. It is $(s,t)$-grid-free when the associated bipartite algebraic graph contains no complete bipart…

Let $H$ be an irreducible hypersurface in $\mathbb{P}^s\times\mathbb{P}^s$, and let $X,Y$ be nonempty Zariski-open subsets of $\mathbb{P}^s$. A hypersurface restriction is $(s,t)$-…

Let $H$ be an irreducible hypersurface in $\mathbb{P}^s\times\mathbb{P}^s$. A set $V\subset\mathbb{P}^s\times\mathbb{P}^s$ is almost-$(s,t)$-grid-free if there are nonempty Zariski…

Let $r\in\mathbb N_0$ and let $K$ be an arbitrary field. The graph $\mathcal G_{r,K}$ has rooted-tree components, and a maximal infinite path is an infinite path starting at the ro…