Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$. Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$. In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$. This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$.

It is easy to see that every k-edge-colouring of the complete graph on $2^k+1$ vertices contains a monochromatic odd cycle. In 1973, Erdős and Graham asked to estimate the smallest L(k) such that every k-edge-colouring of $K_{2^k+1}$ contains a monochromatic odd cycle of length at most L(k). Recently, Girão and Hunter obtained the first nontrivial upper bound by showing that $L(k)=O({2^k}/({k^{1-o(1)}}))$, which improves the trivial bound by a polynomial factor. We obtain an exponential improvement by proving that $L(k)=O(k^{3/2}2^{k/2})$. Our proof combines tools from algebraic combinatorics and approximation theory.

Confirming a conjecture of Erdős on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lovász proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac {n+q-1}{r+q-1}\rfloor$. In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$, we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$-colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon–Frankl–Lovász theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erdős matching conjecture.

A graph G is H-induced-saturated if G is H-free but deleting any edge or adding any edge creates an induced copy of H. There are nontrivial graphs H, such as $P_4$, for which no finite H-induced-saturated graph G exists. We show that for every finite graph H that is not a clique or an independent set, there always exists a countable H-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite H-free graph G such that any graph $G'\ne G$ obtained by making a locally finite set of changes to G contains a copy of H.

Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to apply to hypergraphs in both round-based and lazy settings. We introduce a variant of hypergraph burning that uses an alternative propagation rule for how the fire spreads – if some fixed proportion of vertices are on fire in a hyperedge, then in the next round, the entire hyperedge catches fire.

We obtain bounds on the burning numbers of general hypergraphs, and introduce the concept of the burning distribution, which describes how the burning numbers change as the proportion parameter ranges over $(0,1)$. We also obtain computational results which suggest there is a strong correlation between the automorphism group order and the lazy burning number of a balanced incomplete block design.

This paper resolves the previously open problem of determining the metric dimension of the zero-divisor graph $\Gamma (R)$ for the Boolean ring $R = (\mathbb {Z}_2)^n$. The unique structure of this graph, characterised by its diameter of $3$ and lack of common neighbours, has hindered all standard approaches. We introduce a novel combinatorial method that constructs an explicit resolving set. Consequently, we provide a precise formula for $\dim _m(\Gamma ((\mathbb {Z}_2)^n))$, closing a notable gap in the literature on metric dimensions of zero-divisor graphs. As an application, we compute the metric dimension of a zero-divisor graph of a ring with a Boolean factor.

A trace of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied trace reconstruction problem asks how many traces are required to reconstruct an unknown binary sequence with high probability. In this paper, we study the multidimensional version of this problem for matrices and hypermatrices, where a trace is generated by deleting each row/column of the matrix or each slice of the hypermatrix independently with a constant probability. Previously, Krishnamurthy, Mazumdar, McGregor and Pal showed that $\exp (\widetilde {O}(n^{d/(d+2)}))$ traces suffice to reconstruct any unknown $n\times n$ matrix (for $d=2$) and any unknown $n^{\times d}$ hypermatrix. By developing a dimension reduction procedure and establishing a multivariate version of the Littlewood-type result that lower bounds sparse complex polynomials around $1$, we improve this upper bound by showing that $\exp (\widetilde {O}(n^{3/7}))$ traces suffice to reconstruct any unknown $n\times n$ matrix, and $\exp (\widetilde {O}(n^{3/5}))$ traces suffice to reconstruct any unknown $n^{\times d}$ hypermatrix. In contrast to the earlier bound, our new exponent is bounded away from $1$ even as $d$ becomes very large.

The purpose of this paper is to analyze the degree index and the clustering index in dense random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this paper. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erdős–Rényi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. In addition to the Erdős–Rényi case, we also present a simulation-based analysis for random regular graphs, the Barabási–Albert model, and the Watts–Strogatz model.

We provide four equivalent combinatorial conditions for a simple assembly graph (rigid vertex graph where all vertices are of degree 1 or 4) to have the largest number of Hamiltonian sets of polygonal paths relative to its size. These conditions serve to prove the conjecture that such a maximum, which is equal to $F_{2n+1}-1$, where $F_k$ denotes the $k$th Fibonacci number, is achieved only for special assembly graphs, called tangled cords.

Given a sequence of graphs $G_n$ and a fixed graph H, denote by $T(H, G_n)$ the number of monochromatic copies of the graph H in a uniformly random c-coloring of the vertices of $G_n$. In this paper we study the joint distribution of a finite collection of monochromatic graph counts in networks with multiple layers (multiplex networks). Specifically, given a finite collection of graphs $H_1, H_2, \ldots, H_d$ we derive the joint distribution of $(T(H_1, G_n^{(1)}), T(H_2, G_n^{(2)}), \ldots, T(H_d, G_n^{(d)}))$, where $\mathbf{G}_n = (G_n^{(1)}, G_n^{(2)}, \ldots, G_n^{(d)})$ is a collection of dense graphs on the same vertex set converging in the multiplex cut-metric. The limiting distribution is the sum of two independent components: a multivariate Gaussian and a sum of independent bivariate stochastic integrals. This extends previous results on the marginal convergence of monochromatic subgraphs in a sequence of graphs to the joint convergence of a finite collection of monochromatic subgraphs in a sequence of multiplex networks. Several applications and examples are discussed.

We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\boldsymbol {\Sigma }^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.

In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as noncolourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this noncolourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices.

Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph H in r colours has a sharp threshold in $G_{n,p}$, for all $r \geqslant 2$ and all H in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden’s theorem and Schur’s theorem, also in any number of colours.

A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of $2$ and every power of $2$ appears at most twice. We give three applications of the length generating function for such partitions, denoted by $h_q(n)$. Morier-Genoud and Ovsienko defined the q-analogue of a rational number $[r/s]_q$ in various ways, most of which depend directly or indirectly on the continued fraction expansion of $r/s$. As our first application we show that $[r/s]_q=q\,h_q(n-1)/h_q(n)$ where $r/s$ occurs as the nth entry in the Calkin-Wilf enumeration of the non-negative rationals. Next we consider fence posets which are those which can be obtained from a sequence of chains by alternately pasting together maxima and minima. For every n we show there is a fence poset ${\cal F}(n)$ whose lattice of order ideals is isomorphic to the poset of hyperbinary partitions of n ordered by refinement. For our last application, Morier-Genoud and Ovsienko also showed that $[r/s]_q$ can be computed by taking products of certain matrices which are q-analogues of the standard generators for the special linear group $\operatorname {\mathrm {SL}}(2,{\mathbb Z})$. We express the entries of these products in terms of the polynomials $h_q(n)$.

Free Steiner triple systems (STSs) are infinite structures that are naturally characterised by a universal property. We consider the class of free STSs from a model-theoretic viewpoint. We show that free STSs on any number of generators are elementarily equivalent. We axiomatise their theory and show that it is stable.

Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with K-theory yields a rich combinatorial theory of inhomogeneous deformations, where Schur functions are replaced by their K-analogs, the symmetric Grothendieck functions. We initiate a theory of the Kromatic symmetric function $\overline {X}_G$, a K-theoretic analog of the chromatic symmetric function $X_G$ of a graph G. The Kromatic symmetric function is a generating series for graph colorings in which vertices receive any nonempty set of colors such that neighboring color sets are disjoint. Our main result lifts a theorem of Gasharov (1996), showing that when G is a claw-free incomparability graph, $\overline {X}_G$ is a positive sum of symmetric Grothendieck functions. This suggests a topological interpretation of Gasharov’s theorem. Kromatic symmetric functions of path graphs are not positive in any of several K-analogs of the e-basis, demonstrating that the Stanley–Stembridge conjecture (1993) does not have such a lift to K-theory and so is unlikely to be amenable to a topological perspective. We define a vertex-weighted extension of $\overline {X}_G$ which admits a deletion–contraction relation. Finally, we give a K-analog for $\overline {X}_G$ of the monomial-basis expansion of $X_G$.

We compute the Galois groups of the reductions modulo a prime number p of the generating series of Apéry numbers, Domb numbers and Almkvist–Zudilin numbers. We observe in particular that their behaviour is governed by congruence conditions on p.

The set of sums of two squares plays a significant role in number theory. We establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely on algebraic properties arising from the induced structures on the Stone–Čech compactification of the natural numbers.

Distributed ledgers, including blockchain and other decentralized databases, are designed to store information online where all trusted network members can update the data with transparency. The dynamics of a ledger’s development can be mathematically represented by a directed acyclic graph (DAG). In this paper, we study a DAG model that considers batch arrivals and random delay of attachment. We analyze the asymptotic behavior of this model by letting the arrival rate go to infinity and the inter-arrival time go to zero. We establish that the number of leaves in the DAG, as well as various random variables characterizing the vertices in the DAG, can be approximated by its fluid limit, represented as the solution to a set of delayed partial differential equations. Furthermore, we establish the stable state of this fluid limit and validate our findings through simulations.

One-dependent first passage percolation is a spreading process on a graph where the transmission time through each edge depends on the direct surroundings of the edge. In particular, the classical i.i.d. transmission time $L_{xy}$ is multiplied by $(W_xW_y)^\mu $, a polynomial of the expected degrees $W_x, W_y$ of the endpoints of the edge $xy$, which we call the penalty function. Beyond the Markov case, we also allow any distribution for $L_{xy}$ with regularly varying distribution near $0$. We then run this process on three spatial scale-free random graph models: finite and infinite Geometric Inhomogeneous Random Graphs, including Hyperbolic Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees.

We show that as the penalty function, that is, $\mu $ increases, the transmission time between two far away vertices sweeps through four universal phases: explosive (with tight transmission times), polylogarithmic, polynomial but strictly sublinear, and linear in the Euclidean distance. The strictly polynomial growth phase is a new phenomenon that so far was extremely rare in spatial graph models. All four growth phases are robust in the model parameters and are not restricted to phase boundaries. Further, the transition points between the phases depend nontrivially on the main model parameters: the tail of the degree distribution, a long-range parameter governing the presence of long edges, and the behaviour of the distribution L near $0$. In this paper we develop new methods to prove the upper bounds in all sub-explosive phases. Our companion paper complements these results by providing matching lower bounds in the polynomial and linear regimes.