9 problems

Kojman's conjecture. Closure of the constraint class $\mathcal C$ under homomorphic image should be a key condition for the existence of a universal countable $\mathcal C$-free gra…

Let $\mathcal{F}$ be a class of graphs, and let $\equiv_{\mathcal{F}}$ denote the relation on graphs defined by equality of homomorphism counts from every graph in $\mathcal{F}$. A…

Let $\mathcal{F}$ be a class of graphs. It is homomorphism distinguishing closed if it is maximal among the classes defining its homomorphism indistinguishability relation. A graph…

Let $M'$ and $M$ be Coxeter data belonging to classical series, and let $\operatorname{Hom}_{\mathscr H}(M',M)$ denote the homomorphisms of the corresponding Hecke monoids. A homom…

Let $(W(M),\star)$ and $(W(M'),\star)$ be Hecke monoids of irreducible crystallographic types. A homomorphism is injective parabolic if it is both injective and parabolic, and it i…

Bounded-width boundary conjecture. A homomorphism problem is $\mathbf{\Pi}^1_1$ iff it is not $\mathbf{\Sigma}^1_2$-complete iff it is bounded width.

Classification conjecture. The Exists Right-Factor Problem with fixed algebra $Z$ is in polynomial time for a given variety if and only if the variety has bounded $f$-cores.

Let $G$ be an $n$-vertex, $d$-regular graph, let $K_q$ be the complete graph on $q$ vertices, and let ${\rm hom}(G,K_q)$ denote the number of graph homomorphisms from $G$ to $K_q$.…

Let $\textnormal{And}_k$ be the Andrásfai graph, and let $H$ be an arbitrary homomorphic preimage of $\textnormal{And}_k$. Let $F_k$ be the bipartification defined in Theorem. Homo…