Prescribed Color Patterns in Tight Hamilton Cycles and Spanning Trees

We compare transversal and pattern thresholds for two spanning structures. For every fixed k≥ 3 and $α>0$, we prove that every sufficiently large n-vertex k-graph collection with minimum codegree at least $(1/2+α)n$ contains a tight Hamilton cycle with any prescribed color pattern. Hence the transversal and pattern thresholds for tight Hamilton cycles coincide asymptotically. For spanning trees, we show that for every $α>0$ there exists $c=c(α)>0$ such that every n-vertex graph collection with minimum degree at least $(1/2+α)n$ contains every n-vertex tree with maximum degree at most nᶜ and any prescribed color pattern. On the other hand, we construct spanning trees of maximum degree √ n+1 whose pattern threshold exceeds their transversal threshold by a positive linear term.