Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition with k parts if there exists an ε-regular partition with k parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

In this paper we study a question related to the classical Erdős–Ko–Rado theorem, which states that any family of k-element subsets of the set [n] = {1,. . .,n} in which any two sets intersect has cardinality at most $\binom{n-1}{k-1}$ .

We say that two non-empty families ${\mathcal A}, {\mathcal B}\subset \binom{[n]}{k}$ are s-cross-intersecting if, for any A ∈ ${\mathcal A}$ , B ∈ ${\mathcal B}$ , we have |A ∩ B| ≥ s. In this paper we determine the maximum of | ${\mathcal A}$ |+| ${\mathcal B}$ | for all n. This generalizes a result of Hilton and Milner, who determined the maximum of | ${\mathcal A}$ |+| ${\mathcal B}$ | for non-empty 1-cross-intersecting families.

Let C 6 3 be the 3-uniform hypergraph on {1, . . ., 6} with edges 123,345,561, which can be seen as the analogue of the triangle in 3-uniform hypergraphs. For sufficiently large n divisible by 6, we show that every n-vertex 3-uniform hypergraph H with minimum codegree at least n/3 contains a C 6 3-factor, that is, a spanning subhypergraph consisting of vertex-disjoint copies of C 6 3. The minimum codegree condition is best possible. This improves the asymptotic result obtained by Mycroft and answers a question of Rödl and Ruciński exactly.

In a Markov chain started at a state x, the hitting time τ(y) is the first time that the chain reaches another state y. We study the probability $\mathbb{P}_x(\tau(y) = t)$ that the first visit to y occurs precisely at a given time t. Informally speaking, the event that a new state is visited at a large time t may be considered a ‘surprise’. We prove the following three bounds.

• • In any Markov chain with n states, $\mathbb{P}_x(\tau(y) = t) \le {n}/{t}$ .

• • In a reversible chain with n states, $\mathbb{P}_x(\tau(y) = t) \le {\sqrt{2n}}/{t}$ for $t \ge 4n + 4$ .

• • For random walk on a simple graph with n ≥ 2 vertices, $\mathbb{P}_x(\tau(y) = t) \le 4e \log(n)/t$ .

To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): for random walk on an n-vertex graph, for every initial vertex x,

$$\sum_y \biggl( \sup_{t \ge 0} p^t(x, y) \biggr) = O(\log n). $$

Let $\mathcal F$ ⊂ 2[n] be a family of subsets. The diameter of $\mathcal F$ is the maximum of the size of symmetric differences among pairs of members of $\mathcal F$ . In 1966 Kleitman determined the maximum of | $\mathcal F$ | for fixed diameter. However, this important classical result lacked a characterization of the families meeting the bound. This is remedied in the present paper, where a best possible stability result is established as well.

In Section 4 we introduce a ‘parity trick’ that provides an easy way of deducing the odd case from the even case in both Kleitman's original theorem and its stability version.