An ordered r-matching is an r-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of r-dimensional orders. The theory of ordered 2-matchings is well developed and has connections and applications to extremal and enumerative combinatorics, probability and geometry. On the other hand, in the case $r \ge 3$ much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruciński made some first steps towards a general theory of ordered r-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain.

For a nondegenerate r-graph F, large n, and t in the regime $[0, c_{F} n]$, where $c_F>0$ is a constant depending only on F, we present a general approach for determining the maximum number of edges in an n-vertex r-graph that does not contain $t+1$ vertex-disjoint copies of F. In fact, our method results in a rainbow version of the above result and includes a characterization of the extremal constructions.

Our approach applies to many well-studied hypergraphs (including graphs) such as the edge-critical graphs, the Fano plane, the generalized triangles, hypergraph expansions, the expanded triangles, and hypergraph books. Our results extend old results of Erdős [13], Simonovits [76], and Moon [58] on complete graphs, and can be viewed as a step toward a general density version of the classical Corrádi–Hajnal [10] and Hajnal–Szemerédi [32] theorems.

Our method relies on a novel understanding of the general properties of nondegenerate Turán problems, which we refer to as smoothness and boundedness. These properties are satisfied by a broad class of nondegenerate hypergraphs and appear to be worthy of future exploration.

We give an affirmative answer to a long-standing conjecture of Thomassen, stating that every sufficiently highly connected graph has a k-vertex-connected orientation. We prove that a connectivity of order $O(k^2)$ suffices. As a key tool, we show that for every pair of positive integers d and t, every $(t \cdot h(d))$-connected graph contains t edge-disjoint d-rigid (in particular, d-connected) spanning subgraphs, where $h(d) = 10d(d+1)$. This also implies a positive answer to the conjecture of Kriesell that every sufficiently highly connected graph G contains a spanning tree T such that $G-E(T)$ is k-connected.

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ‘hubs’. We study the higher-order connectivity of such a network by investigating the topological properties of its clique complex. We concentrate on the Betti numbers, a sequence of topological invariants of the complex related to the numbers of holes (equivalently, repeated connections) of different dimensions. We prove that the expected Betti numbers grow sublinearly fast, with the trivial exceptions of those at dimensions 0 and 1. Our result also shows that preferential attachment graphs undergo infinitely many phase transitions within the parameter regime where the limiting degree distribution has an infinite variance. Regarding higher-order connectivity, our result shows that preferential attachment favors higher-order connectivity. We illustrate our theoretical results with simulations.

We prove that, for any finite set of minimal r-graph patterns, there is a finite family $\mathcal F$ of forbidden r-graphs such that the extremal Turán constructions for $\mathcal F$ are precisely the maximum r-graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author [30], where the above statement was established for a single pattern.

We present two applications of this result. First, we construct a finite family $\mathcal F$ of $3$-graphs such that there are exponentially many maximum $\mathcal F$-free $3$-graphs of each large order n and, moreover, the corresponding Turán problem is not finitely stable. Second, we show that there exists a finite family $\mathcal {F}$ of $3$-graphs whose feasible region function attains its maximum on a Cantor-type set of positive Hausdorff dimension.

Ramsey’s theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdős conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of $K_k$, and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko’s conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common and, in fact, no 2-chromatic common graph unsettled for Sidorenko’s conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known.

We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami et al. [Non-three-colourable common graphs exist, Combin. Probab. Comput. 21 (2012), 734–742], and a problem listed by Conlon et al. [Recent developments in graph Ramsey theory, in Surveys in combinatorics 2015, London Mathematical Society Lecture Note Series, vol. 424 (Cambridge University Press, Cambridge, 2015), 49–118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger et al. [Multiplicities of subgraphs, Combinatorica 16 (1996), 123–131] whether there exists a common graph with chromatic number at least four.

Thomassen’s chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph G are the length of its longest cycles and the length of its longest chordless cycles, respectively. Harvey [‘A cycle of maximum order in a graph of high minimum degree has a chord’, Electron. J. Combin. 24(4) (2017), Article no. 4.33, 8 pages] proposed the stronger conjecture: every $2$-connected graph G with minimum degree at least $3$ has $c(G)\geq c'(G)+2$. This conjecture implies Thomassen’s chord conjecture. We observe that wheels are the unique Hamiltonian graphs for which the circumference and the induced circumference differ by exactly one. Thus, we need only consider non-Hamiltonian graphs for Harvey’s conjecture. We propose a conjecture involving wheels that is equivalent to Harvey’s conjecture on non-Hamiltonian graphs. A graph is $\ell $-holed if all its holes have length exactly $\ell $. We prove that Harvey’s conjecture and hence also Thomassen’s conjecture hold for $\ell $-holed graphs and graphs with a small induced circumference.

Geometric parameters in general and curvature in particular play a fundamental role in our understanding of the structure and functioning of real-world networks. Here, the discretisation of the Ricci curvature proposed by Forman is adapted to capture the global influence of the network topology on individual edges of a graph. This is implemented mathematically by assigning communicability distances to edges in the Forman–Ricci definition of curvature. We study analytically both the edge communicability curvature and the global graph curvature and give mathematical characterisations of them. The Forman–Ricci communicability curvature is interpreted ‘physically‘ on the basis of a non-conservative diffusion process taking place on the graph. We then solve analytically a toy model that allows us to understand the fundamental differences between edges with positive and negative Forman–Ricci communicability curvature. We complete the work by analysing three examples of applications of this new graph-theoretic invariant on real-world networks: (i) the network of airport flight connections in the USA, (ii) the neuronal network of the worm Caenorhabditis elegans and (iii) the collaboration network of authors in computational geometry, where we strengthen the many potentials of this new measure for the analysis of complex systems.

We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\mathord{\mathbb R}}^d$. We present combinatorial expressions for the computation of the cumulants and moments of all orders of such subgraph counts, which allow us to estimate the growth of cumulants as the intensity of the underlying Poisson point process goes to infinity. As a consequence, we obtain a central limit theorem with explicit convergence rates under the Kolmogorov distance and connectivity bounds. Numerical examples are presented using a computer code in SageMath for the closed-form computation of cumulants of any order, for any type of connected subgraph, and for any configuration of endpoints in any dimension $d{\geq} 1$. In particular, graph connectivity estimates, Gram–Charlier expansions for density estimation, and correlation estimates for joint subgraph counting are obtained.

Given a fixed small graph H and a larger graph G, an H-factor is a collection of vertex-disjoint subgraphs $H'\subset G$, each isomorphic to H, that cover the vertices of G. If G is the complete graph $K_n$ equipped with independent U(0,1) edge weights, what is the lowest total weight of an H-factor? This problem has previously been considered for $H=K_2$, for example. We show that if H contains a cycle, then the minimum weight is sharply concentrated around some $L_n = \Theta(n^{1-1/d^*})$ (where $d^*$ is the maximum 1-density of any subgraph of H). Some of our results also hold for H-covers, where the copies of H are not required to be vertex-disjoint.

We give a simplified version of the proofs that, outside of their isolated vertices, the complement of the enhanced power graph and of the power graph are connected and have diameter at most $3$.

We study versions of the tree pigeonhole principle, $\mathsf {TT}^1$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics, an outstanding question of which investigation is whether $\mathsf {TT}^1$ is $\Pi ^1_1$-conservative over the ordinary pigeonhole principle, $\mathsf {RT}^1$. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate the analog of this question for Weihrauch reducibility, and give an affirmative answer. In combination with other results, we use this to show that unlike $\mathsf {RT}^1$, the problem $\mathsf {TT}^1$ is not Weihrauch requivalent to any first-order problem. Our proofs develop new combinatorial machinery for constructing and understanding solutions to instances of $\mathsf {TT}^1$.

The triangle removal states that if G contains $\varepsilon n^2$ edge-disjoint triangles, then G contains $\delta (\varepsilon )n^3$ triangles. Unfortunately, there are no sensible bounds on the order of growth of $\delta (\varepsilon )$, and at any rate, it is known that $\delta (\varepsilon )$ is not polynomial in $\varepsilon $. Csaba recently obtained an asymmetric variant of the triangle removal, stating that if G contains $\varepsilon n^2$ edge-disjoint triangles, then G contains $2^{-\operatorname {\mathrm {poly}}(1/\varepsilon )}\cdot n^5$ copies of $C_5$. To this end, he devised a new variant of Szemerédi’s regularity lemma. We obtain the following results:

• • We first give a regularity-free proof of Csaba’s theorem, which improves the number of copies of $C_5$ to the optimal number $\operatorname {\mathrm {poly}}(\varepsilon )\cdot n^5$.

• • We say that H is $K_3$-abundant if every graph containing $\varepsilon n^2$ edge-disjoint triangles has $\operatorname {\mathrm {poly}}(\varepsilon )\cdot n^{\lvert V(H)\rvert }$ copies of H. It is easy to see that a $K_3$-abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are $K_3$-abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative.

Our proofs use a mix of combinatorial, number-theoretic, probabilistic and Ramsey-type arguments.

A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that the contraction poset on pebble trees is isomorphic to the face poset of a convex polytope called pebble tree polytope. Beside providing intriguing generalizations of the classical permutahedra and associahedra, our motivation is that the faces of the pebble tree polytopes provide realizations as convex polytopes of all assocoipahedra constructed by K. Poirier and T. Tradler only as polytopal complexes.

We prove a full measurable version of Vizing’s theorem for bounded degree Borel graphs, that is, we show that every Borel graph $\mathcal {G}$ of degree uniformly bounded by $\Delta \in \mathbb {N}$ defined on a standard probability space $(X,\mu )$ admits a $\mu $-measurable proper edge coloring with $(\Delta +1)$-many colors. This answers a question of Marks [Question 4.9, J. Amer. Math. Soc. 29 (2016)] also stated in Kechris and Marks as a part of [Problem 6.13, survey (2020)], and extends the result of the author and Pikhurko [Adv. Math. 374, (2020)], who derived the same conclusion under the additional assumption that the measure $\mu $ is $\mathcal {G}$-invariant.

In the last two decades the study of random instances of constraint satisfaction problems (CSPs) has flourished across several disciplines, including computer science, mathematics and physics. The diversity of the developed methods, on the rigorous and non-rigorous side, has led to major advances regarding both the theoretical as well as the applied viewpoints. Based on a ceteris paribus approach in terms of the density evolution equations known from statistical physics, we focus on a specific prominent class of regular CSPs, the so-called occupation problems, and in particular on $r$-in-$k$ occupation problems. By now, out of these CSPs only the satisfiability threshold – the largest degree for which the problem admits asymptotically a solution – for the $1$-in-$k$ occupation problem has been rigorously established. Here we determine the satisfiability threshold of the $2$-in-$k$ occupation problem for all $k$. In the proof we exploit the connection of an associated optimization problem regarding the overlap of satisfying assignments to a fixed point problem inspired by belief propagation, a message passing algorithm developed for solving such CSPs.

Let Γ be a finite graph and let $A(\Gamma)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(\Gamma),\mathbb F)$ over an arbitrary field, we construct a natural graph $\Gamma_{\mathcal B}$ from the cup product, called the cohomology basis graph. We show that $\Gamma_{\mathcal B}$ always contains Γ as a subgraph. This provides an effective way to reconstruct the defining graph Γ from the cohomology of $A(\Gamma)$, to characterize the planarity of the defining graph from the algebra of $A(\Gamma)$ and to recover many other natural graph-theoretic invariants. We also investigate the behaviour of the cohomology basis graph under passage to elementary subminors and show that it is not well-behaved under edge contraction.

For a given graph $H$, we say that a graph $G$ has a perfect $H$-subdivision tiling if $G$ contains a collection of vertex-disjoint subdivisions of $H$ covering all vertices of $G.$ Let $\delta _{\mathrm {sub}}(n, H)$ be the smallest integer $k$ such that any $n$-vertex graph $G$ with minimum degree at least $k$ has a perfect $H$-subdivision tiling. For every graph $H$, we asymptotically determined the value of $\delta _{\mathrm {sub}}(n, H)$. More precisely, for every graph $H$ with at least one edge, there is an integer $\mathrm {hcf}_{\xi }(H)$ and a constant $1 \lt \xi ^*(H)\leq 2$ that can be explicitly determined by structural properties of $H$ such that $\delta _{\mathrm {sub}}(n, H) = \left (1 - \frac {1}{\xi ^*(H)} + o(1) \right )n$ holds for all $n$ and $H$ unless $\mathrm {hcf}_{\xi }(H) = 2$ and $n$ is odd. When $\mathrm {hcf}_{\xi }(H) = 2$ and $n$ is odd, then we show that $\delta _{\mathrm {sub}}(n, H) = \left (\frac {1}{2} + o(1) \right )n$.

For each uniformity $k \geq 3$, we construct $k$ uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\Delta$ whose independence polynomial $Z_G$ has a zero $\lambda$ with $\left \vert \lambda \right \vert = O\left (\frac {\log \Delta }{\Delta }\right )$. This disproves a recent conjecture of Galvin, McKinley, Perkins, Sarantis, and Tetali.

We show that if $\Gamma $ is a point group of $\mathbb {R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal {S}$ is a k-pseudomanifold which has a free automorphism of order two, then either $\mathcal {S}$ has a $\Gamma $-symmetric infinitesimally rigid realisation in ${\mathbb R}^{k+1}$ or $k=2$ and $\Gamma $ is a half-turn rotation group. This verifies a conjecture made by Klee, Nevo, Novik and Zheng for the case when $\Gamma $ is a point-inversion group. Our result implies that Stanley’s lower bound theorem for centrally symmetric polytopes extends to pseudomanifolds with a free simplicial automorphism of order 2, thus verifying (the inequality part of) another conjecture of Klee, Nevo, Novik and Zheng. Both results actually apply to a much larger class of simplicial complexes – namely, the circuits of the simplicial matroid. The proof of our rigidity result adapts earlier ideas of Fogelsanger to the setting of symmetric simplicial complexes.