Intersection Problems in Extremal Combinatorics: Theorems, Techniques and Questions Old and New

An efficient container lemma

The non-linear invariance principle of Mossel, O'Donnell and Oleszkiewicz establishes that if f(x1, ..., xn) is a multilinear low-degree polynomial with low influences then the distribution of f(B1, ..., Bn) is close (in various senses) to the distribution of f(G1, ..., Gn), where Bi ∈R {-1, 1} are independent Bernoulli random variables and Gi ~ N(0, 1) are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans--Williamson algorithm for MAXCUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (X1, ..., Xn), (Y1, ..., Yn) such that (i) Matching moments: Xi and Yi have matching first and second moments, (ii) Independence: the variables X1, ..., Xn are independent, as are Y1, ..., Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (X1, ..., Xn) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice ([EQUATION]) which consists of all vectors (x1, ..., xn) ∈ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdos--Ko--Rado theorems) and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, ..., Xn) is the uniform distribution on a slice ([EQUATION]) and (Y1, ..., Yn) consists either of n independent Ber(p) random variables, or of n independent N(p, p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler--Safra structural theorem, and a stability version of the t-intersecting Erdos--Ko--Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem in extremal combinatorics. While for $k=2$ this problem is well understood, due to the work of Tur\'an and of Erd\H{o}s and Stone, only very little is known for $k$-uniform hypergraphs for $k>2$. We focus on the case when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices. Already this very innocent (and maybe somewhat particular looking) problem is still wide open even for $k=3$. We consider a variant of the problem where the large hypergraph $H$ enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H os and S\'os about 30 years ago. We show that every $k$-uniform hypergraph $H$ with density $>2^{1-k}$ with respect to every large collections of $k$-cliques induced by sets of $(k-2)$-tuples contains a copy of $F$. The required density $2^{1-k}$ is best possible as higher order tournament constructions show. Our result can be viewed as a common generalisation of the first extremal result in graph theory due to Mantel (when $k=2$ and the hereditary density condition reduces to a normal density condition) and a recent result of Glebov, Kr\'al', and Volec (when $k=3$ and large subsets of vertices of $H$ induce a subhypergraph of density $>1/4$). Our proof for arbitrary $k\geq 2$ utilises the regularity method for hypergraphs.

On non-strong jumping numbers and density structures of hypergraphs

This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian K-algebras (mainly through the Trace Formula), where K is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite mathbb {Q}-algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null t-designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341–345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21–23, 1983) of the Sauer–Shelah–Perles Lemma, we produce a new one based only in duality in the mathbb {Q}-algebra mathbb {Q}[V_n] of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set [n]:={1,2,ldots , n}. All results are equally true if we replace mathbb {Q}[V_n] by K[V_n], where K is any perfect field of characteristics not =2. The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework.

Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$ vertices and fix a positive integer $k$ such that $1\le k\le r$. A $k$-matching of $\mathcal{H}$ is a collection of edges $\mathcal{M}\subset \mathcal{E}$ such that every subset of $V$ whose cardinality equals $k$ is contained in at most one element of $\mathcal{M}$. The $k$-matching number of $\mathcal{H}$ is the maximum cardinality of a $k$-matching. A well-known problem, posed by Erdős, asks for the maximum number of edges in an $r$-uniform hypergraph under constraints on its $1$-matching number. In this article we investigate the more general problem of determining the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices subject to the constraint that its $k$-matching number is strictly less than $a$. The problem can also be seen as a generalization of the well-known $k$-intersection problem. We propose candidate hypergraphs for the solution of this problem, and show that the extremal hypergraph is among this candidate set when $n\ge 4r\binom{r}{k}^2\cdot a$.

This volume contains eight survey articles by the invited speakers of the 29th British Combinatorial Conference, held at Lancaster University in July 2022. Each article provides an overview of recent developments in a current hot research topic in combinatorics. These topics span graphs and hypergraphs, Latin squares, linear programming, finite fields, extremal combinatorics, Ramsey theory, graph minors and tropical geometry. The authors are among the world's foremost researchers on their respective topics but their surveys are aimed at nonspecialist readers: they are written clearly with little prior knowledge assumed and with pointers to the wider literature. Taken together these surveys give a snapshot of the research frontier in contemporary combinatorics, making the latest developments accessible to researchers and graduate students in mathematics and theoretical computer science with an interest in combinatorics and helping them to keep abreast of the field.

Graphons are analytic objects associated with convergent sequences of dense graphs. Finitely forcible graphons, that is, those determined by finitely many subgraph densities, are of particular interest because of their relation to various problems in extremal combinatorics and theoretical computer science. Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon always has finite dimension, which would have implications on the minimum number of parts in its weak ε-regular partition. We disprove the conjecture by constructing a finitely forcible graphon with the space of typical vertices that has infinite dimension.

Let $G$ be a cancellative $3$-uniform hypergraph in which the symmetric difference of any two edges is not contained in a third one. Equivalently, a $3$-uniform hypergraph $G$ is cancellative if and only if $G$ is $\{F_4, F_5\}$-free, where $F_4 = \{abc, abd, bcd\}$ and $F_5 = \{abc, abd, cde\}$. A classical result in extremal combinatorics stated that the maximum size of a cancellative hypergraph is achieved by the balanced complete tripartite $3$-uniform hypergraph, which was firstly proved by Bollobás and later by Keevash and Mubayi. In this paper, we consider spectral extremal problems for cancellative hypergraphs. More precisely, we determine the maximum $p$-spectral radius of cancellative $3$-uniform hypergraphs, and characterize the extremal hypergraph. As a by-product, we give an alternative proof of Bollobás' result from spectral viewpoint.

On local Turán problems

Let f(n,v,e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e≥3, what is the smallest integer d=d(e) such that f(n,e+d,e)=o(n2)? This question has its origins in work of Brown, Erdős and Sós from the early 70's and the standard conjecture is that d(e)=3 for every e≥3. The state of the art result regarding this problem was obtained in 2004 by Sárközy and Selkow, who showed that f(n,e+2+⌊log2⁡e⌋,e)=o(n2). The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for d(10) from 5 to 4. We obtain the first asymptotic improvement over the Sárközy–Selkow bound, showing thatf(n,e+O(log⁡e/log⁡log⁡e),e)=o(n2).

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.

We consider a generalisation of the classic combinatorial problem of P. Erdős and A. Hajnal in the theory of hypergraphs to the case of prescribed colourings. We investigate the value mpr(n, r) equal to the minimum number of edges of a hypergraph in the class of n-uniform hypergraphs with prescribed chromatic number greater than r. We obtain a lower bound for this value which is better than the known results for r ≥ 3. Moreover, we give a sufficient conditions for existence of a prescribed r-colourability of an n-uniform hypergraph in terms of restrictions on the intersections of edges. As a corollary we obtain a new bound for the characteristic equal to the minimum number of edges of a hypergraph in the class of n-uniform simple hypergraphs (in which any two edges have at most one common vertex) with the prescribed chromatic number greater than r.

The non-linear invariance principle of Mossel, O’Donnell, and Oleszkiewicz establishes that if f ( x 1 ,… , x n ) is a multilinear low-degree polynomial with low influences, then the distribution of if f ( b 1 ,…, b n ) is close (in various senses) to the distribution of f ( G 1 ,…, G n ), where B i ∈ R {-1,1} are independent Bernoulli random variables and G i ∼ N(0,1) are independent standard Gaussians. The invariance principle has seen many applications in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans–Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO’s invariance principle works for any two vectors of hypercontractive random variables ( X 1 ,… , X n ),( Y 1 ,… , Y n ) such that (i) Matching moments : X i and Y i have matching first and second moments and (ii) Independence : the variables X 1 ,… , X n are independent, as are Y 1 ,…, Y n . The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions X 1 ,… , X n in which the individual coordinates are not independent. A common example is the uniform distribution on the slice ( [ n ] k ) which consists of all vectors ( x 1 ,…, x n )∈{0,1} n with Hamming weight k . The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdős–Ko–Rado theorems), and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which ( X 1 ,…, X n ) is the uniform distribution on a slice ( [ n ] pn and ( Y 1 ,… , Y n ) consists either of n independent Ber( p ) random variables, or of n independent N( p , p (1- p )) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain’s tail theorem, a version of the Kindler–Safra structural theorem, and a stability version of the t -intersecting Erdős–Ko–Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra, and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

This is largely an expository paper, revisiting some ideas about compact 2-categories, in which each 1-cell has both a left and a right adjoint. In the special case with only one 0-cell (where the 1-cells are usually called “objects”) we obtain a compact strictly monoidal category. Assuming furthermore that the 2-cells describe a partial order, we obtain a compact partially ordered monoid, which has been called a pregroup. Indeed, a pregroup in which the left and right adjoints coincide is just a partially ordered group (= pogroup).A brief exposition of recent joint work with Preller and Lambek “Mathematical Structures in Computer Science”, 17, (2007) will be given here, investigating free compact strictly monoidal categories, which may be said to describe computations in pregroups. Free pregroups lend themselves to the study of grammar in natural languages such as English. While one would not expect to find a connection between linguistics and physics, applications of (free) compact symmetric monoidal categories to physics have been made by Coecke “The Logic of Entanglement” (2004), Abramsky and Coecke “Proceedings of 19th IEEE Conference on Logic in Computer Science”, pp. 415–425 (2004), Abramsky and Duncan “Mathematical Structures in Computer Science”, 16, 469–489 (2006), Selinger “Electronic Notes in Theoretical Computer Science”, 170, 139–163 (2007).Compact symmetric monoidal categories had already been studied by Kelly and Laplaza “Journal of Pure and Applied Algebra”, 19, 193–213 (1980), who called them “compact closed” and by Barr “Lecture Notes in Mathematics”, 752 (1979), “Journal of Pure and Applied Algebra”, 111, 1–20 (1996), “Theoretical Computer Science”, 139, 115–130 (1995), who called them “compact star-autonomous”. I had intended to show that Feynman diagrams for quantum electro-dynamics (QED) could be described by certain compact Barr-autonomous categories, but was disappointed to find that these reduced to a rather degenerate case, that of partially ordered groups (= pogroups). Still, I will reluctantly present an extension of this idea from QED to the Standard Model. Finally, I will briefly review the transition from 2-categories to the bicategories of Bénabou “Lecture Notes in Mathematics” 47, 1–77 (1967), using methods of Bourbaki “Algebre multilineaire” (1948) and Gentzen (see Kleene “Introduction to metamathematics” (1952)), which may also be of interest in physics.KeywordsDivision RingMonoidal CategoryLeft AdjointGeneralize ContractionAdjoint PairThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.