Abstract. We study conditions under which an edge-colored hypergraph has a particular substructure that contains more than the trivially guaranteed number of monochromatic edges. Our main result solves this problem for perfect matchings under minimum degree conditions. This answers recent questions of Gishboliner, Glock, and Sgueglia and of Balogh, Treglown, and Zárate-Guerén.

Abstract. Given graphs [Formula: see text] and [Formula: see text], a perfect [Formula: see text]-tiling in [Formula: see text] is a collection of vertex-disjoint copies of [Formula: see text] in [Formula: see text] that together cover all the vertices in [Formula: see text]. The study of the minimum degree threshold forcing a perfect [Formula: see text]-tiling in a graph [Formula: see text] has a long history, culminating in the Kühn–Osthus theorem [D. Kühn and D. Osthus, Combinatorica, 29 (2009), pp. 65–107] which resolves this problem, up to an additive constant, for all graphs [Formula: see text]. In this paper we initiate the study of the analogous question for edge-ordered graphs. In particular, we characterize for which edge-ordered graphs [Formula: see text] this problem is well-defined. We also apply the absorbing method to asymptotically determine the minimum degree threshold for forcing a perfect [Formula: see text]-tiling in an edge-ordered graph, where [Formula: see text] is any fixed monotone path.

Abstract. A result of Balogh et al. yields the minimum degree threshold that ensures a 2-colored graph contains a perfect matching of significant color-bias (i.e., a perfect matching that contains significantly more than half of its edges in one color). In this note we prove an analogous result for perfect matchings in [Formula: see text]-uniform hypergraphs. More precisely, for each [Formula: see text] and [Formula: see text] we determine the minimum [Formula: see text]-degree threshold for forcing a perfect matching of significant color-bias in an [Formula: see text]-colored [Formula: see text]-uniform hypergraph.

Let $\vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $\vec{T}_4$'s, and this bound is asymptotically tight. We also improve the best known bound on the minimum total degree for partitioning oriented graphs into vertex disjoint $\vec{T}_k$'s.

Balogh, Csaba, Jing, and Pluhár [ Electron. J. Combin., 27 (2020)] recently determined the minimum degree threshold that ensures a 2-colored graph $G$ contains a Hamilton cycle of significant color bias (i.e., a Hamilton cycle that contains significantly more than half of its edges in one color). In this short note we extend this result, determining the corresponding threshold for $r$-colorings.

Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris, and Samotij [ J. Amer. Math. Soc., 28 (2015), pp. 669--709], and independently Saxton and Thomason [ Invent. Math., 201 (2015), pp. 925--992], developed very general container theorems for independent sets in hypergraphs, both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán) properties of (hyper)graphs and the integers. In particular we do the following: (a) We generalize the random Ramsey theorem of Rödl and Ruciński [ Combinatorics, Paul Erdös Is Eighty, Vol. 1, Bolyai Soc. Math. Stud., János Bolyai Mathematical Society, Budapest, 1993, pp. 317--346; Random Structures Algorithms, 5 (1994), pp. 253--270; J. Amer. Math. Soc., 8 (1995), pp. 917--942] by providing a resilience analogue. Our result unifies and generalizes several fundamental results in the area including the random version of Turán's theorem due to Conlon and Gowers [ Ann. of Math., 184 (2016), pp. 367--454] and Schacht [ Ann. of Math., 184 (2016), pp. 331--363]. (b) The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter [ Random Structures Algorithms, 11 (1997), pp. 245--276]. (c) All of the above results in fact hold for uniform hypergraphs. (d) For a (hyper)graph $H$, we determine, up to an error term in the exponent, the number of $n$-vertex (hyper)graphs $G$ that have the Ramsey property with respect to $H$ (that is, whenever $G$ is $r$-colored, there is a monochromatic copy of $H$ in $G$). (e) We strengthen the random Rado theorem of Friedgut, Rödl, and Schacht [ Random Structures Algorithms, 37 (2010), pp. 407--436] by proving a resilience version of the result. (f) For partition regular matrices $A$ we determine, up to an error term in the exponent, the number of subsets of $\{1,\dots,n\}$ for which there exists an $r$-coloring which contains no monochromatic solutions to $Ax=0$. Along the way a number of open problems are posed.

An important result of Komlós [Tiling Turán theorems, Combinatorica, 2000] yields the asymptotically exact minimum degree threshold that ensures a graph $G$ contains an $H$-tiling covering an $x$th proportion of the vertices of $G$ (for any fixed $x \in (0,1)$ and graph $H$). We give a degree sequence strengthening of this result which allows for a large proportion of the vertices in the host graph $G$ to have degree substantially smaller than that required by Komlós's theorem. We also demonstrate that for certain graphs $H$, the degree sequence condition is essentially best possible in more than one sense.

Given a linear equation L , a set A of integers is L -free if A does not contain any ‘non-trivial’ solutions to L . This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving L -free sets of integers. The main questions we consider involve deciding whether a finite set of integers A has an L -free subset of a given size, and counting all such L -free subsets. We also raise a number of open problems.

A famous conjecture of Pósa from 1962 asserts that every graph on $n$ vertices and with minimum degree at least $2n/3$ contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Komlós, Sárközy, and Szemerédi [Random Structures Algorithms, 9 (1996) pp. 193--211]. In this paper we prove a degree sequence version of Pósa's conjecture: Given any $\eta >0$, every graph $G$ of sufficiently large order $n$ contains the square of a Hamilton cycle if its degree sequence $d_1\leq \dots \leq d_n$ satisfies $d_i \geq (1/3+\eta)n+i$ for all $i \leq n/3$. The degree sequence condition here is asymptotically best possible. Our approach uses a hybrid of the regularity-blowup method and the connecting-absorbing method.

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let $\delta_{\rm Ore}(H,n)$ be the smallest number k such that every graph G whose order n is divisible by $|H|$ and with $d(x)+d(y)\geq k$ for all nonadjacent $x\not=y\in V(G)$ contains a perfect H-packing. We determine $\lim_{n\to\infty}\delta_{\rm Ore}(H,n)/n$.