Transversal factors and spanning trees

Since the proof of a "colorful" version [Caratheodory's theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_%28convex_hull%29) by Bárány in 1982, it has been an important problem to obtain colorful extensions other classical results discrete geometry (for instance Tverberg's theorem). The present paper continues this line research, but context extremal graph theory rather than geometry. Mantel's theorem from 1907 states that every $n$-vertex on more $n^2/4$ edges contains triangle. In [Ron Aharoni, Matt DeVos, Sebastián González Hermosillo de la Maza, Amanda Montejano, and Robert Šámal, A rainbow Mantel’s Theorem, Advances Combinatorics 2020:2, 12 pp](https://arxiv.org/abs/1812.11872v2), "rainbow", "colored", or variant was considered : given three graphs $G_1,G_2,G_3$ same vertex set size $n$, what average degree conditions force existence "rainbow triangle" (a triangle $\{e_1,e_2,e_3\}$ such each edge $e_i$ belongs $G_i$)? By taking copies $G$ we see colored is at least as hard original problem, cited above provided construction showing case strictly harder problem. It suggested study minimum thresholds for variants problems combinatorics, Dirac's (every $n/2$ Hamiltonian cycle). particular, Joos Kim proved 2020 condition guarantees $n$-cycle: namely if are $n$ vertices, then there $n$-cycle comprising one graph. follow research. two major extended setting here Kühn Osthus sharp perfect packing any $F$, generalizing Hajnal-Szemerédi theorem), Komlós, Sárközy Szemerédi contain spanning tree without large vertices). Amazingly, (stronger) versions conditions.