We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as normal sets. We show that for any $\epsilon>0$, there exists $n_0$ such that if $n>n_0$ and A is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon }}$, then $A^2 \supseteq A_n$. This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$.

Our proof strategy combines two types of techniques. The first is ‘traditional’ techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck–Shalev, Larsen–Shalev, and more recently, Larsen–Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$-colouring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$. We prove a result which says that if one replaces $K_n^k$ in Gyárfás’ theorem by any ‘expansive’ $k$-uniform hypergraph on $n$ vertices (that is, a $k$-uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.

Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$-partite $r$-uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$, there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$, then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$). We prove our results in this dual setting.

For a graph $H$ and a hypercube $Q_n$, $\textrm{ex}(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $\lim _{n \rightarrow \infty } \textrm{ex}(Q_n, H)/|E(Q_n)| \gt 0$, $H$ is said to have a positive Turán density in a hypercube or simply a positive Turán density; otherwise, it has zero Turán density. Determining $\textrm{ex}(Q_n, H)$ and even identifying whether $H$ has a positive or zero Turán density remains a widely open question for general $H$. By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of graphs, ones having so-called partite representation, that have zero Turán density. He asked whether this gives a characterisation, that is, whether a graph has zero Turán density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of graphs which have no partite representation, but on the other hand, have zero Turán density. In addition, we show that any graph whose every block has partite representation has zero Turán density in a hypercube.

Let $\mathcal{F}$ be an intersecting family. A $(k-1)$-set $E$ is called a unique shadow if it is contained in exactly one member of $\mathcal{F}$. Let ${\mathcal{A}}=\{A\in \binom{[n]}{k}\colon |A\cap \{1,2,3\}|\geq 2\}$. In the present paper, we show that for $n\geq 28k$, $\mathcal{A}$ is the unique family attaining the maximum size among all intersecting families without unique shadow. Several other results of a similar flavour are established as well.

We show that an $n$-uniform maximal intersecting family has size at most $e^{-n^{0.5+o(1)}}n^n$. This improves a recent bound by Frankl ((2019) Comb. Probab. Comput. 28(5) 733–739.). The Spread Lemma of Alweiss et al. ((2020) Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing.) plays an important role in the proof.

The notion of cross-intersecting set pair system of size $m$ , $ (\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m )$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne \emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le\binom{a+b}{a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$ . Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$ . Such a system is called $1$ -cross-intersecting. We show that these systems are related to perfect graphs, clique partitions of graphs, and finite geometries. We prove that their maximum size is

• at least $5^{n/2}$ for $n$ even, $a=b=n$ ,

• equal to $\bigl (\lfloor \frac{n}{2}\rfloor +1\bigr )\bigl (\lceil \frac{n}{2}\rceil +1\bigr )$ if $a=2$ and $b=n\ge 4$ ,

• at most $|\cup _{i=1}^m A_i|$ ,

• asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ( $|A_i\cap A_j|\le 1$ for $i\ne j$ ),

• asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.

For a graph G and a family of graphs $\mathcal {F}$, the Turán number ${\mathrm {ex}}(G,\mathcal {F})$ is the maximum number of edges an $\mathcal {F}$-free subgraph of G can have. We prove that ${\mathrm {ex}}(G,\mathcal {F})\ge {\mathrm {ex}}(K_r, \mathcal {F})$ if the chromatic number of G is r and $\mathcal {F}$ is a family of connected graphs. This result answers a question raised by Briggs and Cox [‘Inverting the Turán problem’, Discrete Math. 342(7) (2019), 1865–1884] about the inverse Turán number for all connected graphs.

A hypergraph $\mathcal{F}$ is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of $\mathcal{F}$ . Mubayi and Verstraëte showed that for every $k \ge d+1 \ge 3$ and $n \ge (d+1)k/d$ every $k$ -graph $\mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $\binom{n-1}{k-1}$ edges. They conjectured that the same conclusion holds for all $d \ge k \ge 4$ and sufficiently large $n$ . We confirm their conjecture by proving a stronger statement.

They also conjectured that for $m \ge 4$ and sufficiently large $n$ the maximum size of a $3$ -graph on $n$ vertices without a non-trivial intersecting subgraph of size $3m+1$ is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.

Suppose G and H are bipartite graphs and $L: V(G)\to 2^{V(H)}$ induces a partition of $V(H)$ such that the subgraph of H induced between $L(v)$ and $L(v')$ is a matching, whenever $vv'\in E(G)$ . We show for each $\varepsilon>0$ that if H has maximum degree D and $|L(v)| \ge (1+\varepsilon )D/\log D$ for all $v\in V(G)$ , then H admits an independent transversal with respect to L, provided D is sufficiently large. This bound on the part sizes is asymptotically sharp up to a factor $2$ . We also show some asymmetric variants of this result.

A family of vectors in [k]n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k]n invariant under a transitive group of symmetries is o(kn), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.

The Corners theorem states that for any α > 0 there exists an N0 such that for any abelian group G with |G| = N ≥ N0 and any subset A ⊂ G×G with |A| ≥ αN2 we can find a corner in A, i.e. there exist x, y, d ∈ G with d ≠ 0 such that (x,y),(x+d,y),(x,y+d) ∈ A.

Here, we consider a stronger version, in which we try to find many corners of the same size. Given such a group G and subset A, for each d ∈ G we define Sd={(x,y) ∈ G × G: (x,y),(x+d,y),(x,y+d) ∈ A}. So |Sd| is the number of corners of size d. Is it true that, provided N is sufficiently large, there must exist some d ∈G\{0} such that |Sd|>(α3-ϵ)N2?

We answer this question in the negative. We do this by relating the problem to a much simpler-looking problem about random variables. Then, using this link, we show that there are sets A with |Sd|>Cα3.13N2 for all d ≠ 0, where C is an absolute constant. We also show that in the special case where $G = {\mathbb{F}}_2^n$, one can always find a d with |Sd|>(α4-ϵ)N2.

We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry-quasirandom 3-graphs.

Our first result asserts that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α(n – 2) have a tight Hamilton cycle.

Our second result concerns the minimum 1-degree sufficient for such 3-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every d, α > 0 satisfying d + α > 1, any sufficiently large n-vertex such 3-graph H of density d and minimum 1-degree at least $\alpha \left({\matrix{{n - 1} \cr 2 \cr } } \right)$ has a tight Hamilton cycle.

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braß–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).

In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).

Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).

On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.

A family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any $$k \le ({1 \over 2} - o(1))n$$, if X is an n-element set, and $${\cal F} = {\cal F}_1 \cup {\cal F}_2 \cup \cdots \cup {\cal F}_r $$, where each $$ {\cal F}_i $$ is an intersecting family of k-element subsets of X, then $$|{\cal F}| \le \left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right)$$, with equality only if $${\cal F} = \{ S \subset X:|S| = k,\;S \cap R \ne \emptyset \} $$ for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $$k < (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit-length circular track, consider $m$ runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning at a distance at least $1/m$ from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting.

A family of subsets of {1,. . .,n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1,. . .,n}. In this paper we study the following problem: How many intersecting families of k-subsets of {1,. . .,n} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for n ≥ 2k+2+2$\sqrt{k\log k}$ and k → ∞. Moreover, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.

We prove that Boolean functions on $S_{n}$, whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of $n$ whose largest part has size at least $n-t$, are close to being unions of cosets of stabilizers of $t$-tuples. We also obtain an edge-isoperimetric inequality for the transposition graph on $S_{n}$ which is asymptotically sharp for subsets of $S_{n}$ of size $n!/\text{poly}(n)$, using eigenvalue techniques. We then combine these two results to obtain a sharp edge-isoperimetric inequality for subsets of $S_{n}$ of size $(n-t)!$, where $n$ is large compared to $t$, confirming a conjecture of Ben Efraim in these cases.

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.