A seminal result of Hajnal and Szemerédi states that if a graph G with n vertices has minimum degree δ(G) ≥ (r − 1)n/r for some integer r ≥ 2, then G contains a K r -factor, assuming r divides n. Extremal examples which show optimality of the bound on δ(G) are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Túran theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results:• For any r > ℓ ≥ 2, if G is a graph satisfying δ(G) ≥ r−ℓ r−ℓ+1 n+Ω(n) and α ℓ (G) = o(n), that is, a largest K ℓ -free induced subgraph has at most o(n) vertices, then G contains a K r -factor. This is optimal for ℓ = r − 1 and extends a result of Balogh, Molla, and Sharifzadeh who considered the case r = 3.• If a graph G satisfies δ(G) = Ω(n) and α * r (G) = o(n), that is, every induced K r -free r-partite subgraph of G has at least one vertex class of size o(n), then it contains a K r -factor. A similar statement is proven for a general graph

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

It is known that a sequence {Π i } i∈N of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Π i converges to 1∕24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Π i converges to |S|∕24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.

We prove the following 30-year old conjecture of Győri and Tuza: the edges of every n-vertex graph G can be decomposed into complete graphs C 1 , . . . , C ℓ of orders two and three such that |C 1 | + · · · + |C ℓ | ≤ (1/2 + o(1))n 2 . This result implies the asymptotic version of the old result of Erdős, Goodman and Pósa that asserts the existence of such a decomposition with ℓ ≤ n 2 /4.