11 problems

For an integer $r\geq 5$, let $C_r$ denote the cycle on $r$ vertices, and let $rsat(n,C_r)$ be its rainbow saturation number. Asymptotic rainbow saturation conjecture for cycles. T…

Let $C_5$ denote the cycle on five vertices, and let $rsat(n,C_5)$ be its rainbow saturation number. Eventual exact rainbow saturation conjecture for $C_5$. There exists…

Let $F$ be a nonempty graph, and let $rsat(n,F)$ denote the minimum number of edges in an $F$-rainbow saturated graph of order $n$. Behague–Johnston–Letzter–Morrison–Ogden's conjec…

Let $H$ be a graph, and let $\operatorname{prsat}'(n,H)$ denote the BJR proper rainbow saturation number: the minimum number of edges in a graph admitting a proper edge-colouring w…

Let $K_r$ be the complete graph on $r$ vertices, and let $\operatorname{rsat}(n,K_r)$ denote its rainbow saturation number. Girao, Lewis, and Popielarz's conjecture. For every…

Let $H$ be a non-empty graph, and let $\operatorname{rsat}(n,H)$ denote the minimum number of edges in an $H$-rainbow saturated graph on $n$ vertices, where an edge-coloured graph…

Rainbow hypergraph saturation conjecture. For every $k$-uniform hypergraph $H$, we have

Linear rainbow saturation conjecture. For every graph $H$ there is a constant $c=c(H)$ such that

Let $H$ be any graph, and let $\operatorname{sat}\left(n,\mathfrak{R}(H)\right)$ denote its rainbow saturation number when the colour palette is unrestricted. Universal linear rain…

Let $H$ be a connected graph other than an odd clique with a rotated edge. Assume that $H$ has an edge not contained in a triangle and has no conical vertex. Linear rainbow saturat…

Let $K_r$ be the complete graph on $r$ vertices, and let $\operatorname{sat}_{t}\left(n,\mathfrak{R}(K_r)\right)$ denote the minimum number of edges in an $n$-vertex graph whose ed…