We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least

\begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*}

$\exp\!(n\log n-\Theta (n))$

$\ell$

$(k-\ell )\mid n$

$(k-\ell )\mid k$

$(k-\ell )\nmid k$

$\ell \lt k/2$

A set $S\subset {\mathbb {N}}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$, $s_1\leqslant s_2$) are distinct. A set $S\subset {\mathbb {N}}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$. In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $\mathbb {F}_q[t]$-analogue of Montgomery's conjecture for convolutions of the von Mangoldt function.

Let G be a graph with m edges, minimum degree $\delta $ and containing no cycle of length 4. Answering a question of Bollobás and Scott, Fan et al. [‘Bisections of graphs without short cycles’, Combinatorics, Probability and Computing 27(1) (2018), 44–59] showed that if (i) G is $2$-connected, or (ii) $\delta \ge 3$, or (iii) $\delta \ge 2$ and the girth of G is at least 5, then G admits a bisection such that $\max \{e(V_1),e(V_2)\}\le (1/4+o(1))m$, where $e(V_i)$ denotes the number of edges of G with both ends in $V_i$. Let $s\ge 2$ be an integer. In this note, we prove that if $\delta \ge 2s-1$ and G contains no $K_{2,s}$ as a subgraph, then G admits a bisection such that $\max \{e(V_1),e(V_2)\}\le (1/4+o(1))m$.

A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every fixed $C$, if each vertex $v \in \bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$, then the (list) chromatic number of $\bigcup _{i=1}^m G_i$ is at most $D + o(D)$. This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy and Postle, and we derive this result from a more general list colouring result in the setting of ‘colour degrees’ that also implies a result of Reed and Sudakov.

We show that for every $n\in \mathbb N$ and $\log n\le d\lt n$, if a graph $G$ has $N=\Theta (dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every $n$-vertex $d$-regular graph.

A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum _{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets A. In 1986, he asked if this bound is attained for the set of prime numbers. In this article, we answer in the affirmative.

As further applications of the method, we make progress towards a question of Erdős, Sárközy and Szemerédi from 1968. We also refine the classical Davenport–Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy and Szemerédi from 1966.

We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$. The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and Füredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k\geq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-\left\lfloor{\frac{k}{2^{n-d}}}\right\rfloor$, while for $n \gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$, and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.

An old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$ -vertex graph are of order $O(\!\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ . We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\!\log n)$ , then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ .

Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$. We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.

In 1999, Jacobson and Lehel conjectured that, for $k \geq 3$ , every k-regular Hamiltonian graph has cycles of $\Theta (n)$ many different lengths. This was further strengthened by Verstraëte, who asked whether the regularity can be replaced with the weaker condition that the minimum degree is at least $3$ . Despite attention from various researchers, until now, the best partial result towards both of these conjectures was a $\sqrt {n}$ lower bound on the number of cycle lengths. We resolve these conjectures asymptotically by showing that the number of cycle lengths is at least $n^{1-o(1)}$ .

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists M such that every triangle-free graph G has an $\epsilon$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $\epsilon$ -approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon \left\lvert{V(G)}\right\rvert^2$ edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.

We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.

The Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1’s in an m × n matrix with all entries 0 or 1 containing no s × t submatrix consisting entirely of 1’s. We show that a classical upper bound for z(m, n; s, t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.

The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

Let A be a finite set with , let n be a positive integer, and let $A^n$ denote the discrete $n\text {-dimensional}$ hypercube (that is, $A^n$ is the Cartesian product of n many copies of A). Given a family $\langle D_t:t\in A^n\rangle $ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle $ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild ‘stationarity’ condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales-Jewett theorem.

We prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.

We answer four questions from a recent paper of Rao and Shinkar [17] on Lipschitz bijections between functions from {0, 1}n to {0, 1}. (1) We show that there is no O(1)-bi-Lipschitz bijection from Dictator to XOR such that each output bit depends on O(1) input bits. (2) We give a construction for a mapping from XOR to Majority which has average stretch $O(\sqrt{n})$ , matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $\phi \colon \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $${\rm{XOR }}(x) = {\rm{ Majority }}(\phi (x))$$ for all $x \in \{0,1\}^n$ . (4) We show that with high probability there is an O(1)-bi-Lipschitz mapping from Dictator to a uniformly random balanced function.

Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then

$$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$

In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then

$$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$

$d \rightarrow \infty$

$$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$

Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs $K_{d,d}$ , then ${\gamma_\circ}(G) = \frac{n}{2}$ . We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that ${\gamma_\circ}(G) \sim {n}/{2}$ as $d \rightarrow \infty$ . Therefore both the girth and regularity conditions are required for the main result.

In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.