Graphs of low average degree without independent transversals

An independent transversal of a graph G $G$ with vertex partition P ${\mathscr{P}}$ is an set intersecting each block in single vertex. Wanless and Wood proved that if has size at least t $t$ the average degree vertices most ∕ 4 $t\unicode{x02215}4$ , then exists. We present construction showing this result optimal: for any ε > 0 $\varepsilon \gt 0$ sufficiently large there forest into parts such ( 1 + ) $(\frac{1}{4}+\varepsilon )t$ no transversal. This unexpectedly shows methods related to entropy compression as Rosenfeld–Wanless–Wood scheme or Local Cut Lemma are tight problem. Further constructions given variants problem, including hypergraph version.