On tt-Intersecting Families of Permutations

We prove that there exists a constant $c_0$ such that for any $t \in \mathbb{N}$ and any $n\geq c_0 t$, if $A \subset S_n$ is a $t$-intersecting family of permutations then$|A|\leq (n-t)!$. Furthermore, if $|A|\ge 0.75(n-t)!$ then there exist $i_1,\ldots,i_t$ and $j_1,\ldots,j_t$ such that $\sigma(i_1)=j_1,\ldots,\sigma(i_t)=j_t$ holds for any $\sigma \in A$. This shows that the conjectures of Deza and Frankl (1977) and of Cameron (1988) on $t$-intersecting families of permutations hold for all $t \leq c_0 n$. Our proof method, based on hypercontractivity for global functions, does not use the specific structure of permutations, and applies in general to $t$-intersecting sub-families of `pseudorandom' families in $\{1,2,\ldots,n\}^n$, like $S_n$

A Hilton–Milner-type theorem and an intersection conjecture for signed sets

A family A of sets is said to be intersecting if any two sets in A intersect (i.e. have at least one common element). A is said to be centred if there is an element common to all the sets in A; otherwise, A is said to be non-centred. For any r ∈ [n] := {1, . . . , n} and any integer k ≥ 2, let Sn,r,k be the family {{(x1, y1), . . . , (xr, yr)}: x1, . . . , xr are distinct elements of [n], y1, . . . , yr ∈ [k]} of k-signed r-sets on [n]. Let m := max{0, 2r−n}.We establish the following Hilton–Milner-type theorems, the second of which is proved using the first: (i) If A1 and A2 are non-empty cross-intersecting (i.e. any set in A1 intersects any set in A2) sub-families of Sn,r,k, then |A1| + |A2| ≤ n R K r −r i=m r I (k − 1) I n – r r – I K r−i + 1. (ii) If A is a non-centred intersecting sub-family of Sn,r,k, 2 ≤ r ≤ n, then |A| ≤ n – 1 r – 1 K r−1 −r−1 i=m r I (k − 1) I n − 1 – r r − 1 – I K r−1−i + 1 if r < n; k r−1 − (k − 1) r−1 + k − 1 if r = n. We also determine the extremal structures. (ii) is a stability theorem that extends Erdős–Ko–Rado-type results proved by various authors. We then show that (ii) leads to further evidence for an intersection conjecture suggested by the author about general signed set systems.peer-reviewe

Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs

In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs. Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber. Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses. Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture

High dimensional Hoffman bound and applications in extremal combinatorics

One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound is sharp on the tensor power of a graph whenever it is sharp for the original graph. In this paper, we introduce the related problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many of the prominent open problems in extremal combinatorics, such as the Tur\'an problem for (hyper-)graphs, can be encoded as special cases of this problem. We also give a new generalization of the Hoffman bound for hypergraphs which is sharp for the tensor power of a hypergraph whenever it is sharp for the original hypergraph. As an application of our Hoffman bound, we make progress on the problem of Frankl on families of sets without extended triangles from 1990. We show that if $\frac{1}{2}n\le2k\le\frac{2}{3}n,$ then the extremal family is the star, i.e. the family of all sets that contains a given element. This covers the entire range in which the star is extremal. As another application, we provide spectral proofs for Mantel's theorem on triangle-free graphs and for Frankl-Tokushige theorem on $k$-wise intersecting families

On the union of intersecting families

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s 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 $r \geq 2$ and for any $k \leq (\tfrac{1}{2}-o(1))n$, if $X$ is an $n$-element set, and $\mathcal{F} = \mathcal{F}_1 \cup \mathcal{F}_2 \cup \ldots \cup \mathcal{F}_r$, where each $\mathcal{F}_i$ is an intersecting family of $k$-element subsets of $X$, then $|\mathcal{F}| \leq {n \choose k} - {n-r \choose k}$, with equality only if $\mathcal{F} = \{S \subset X:\ |S|=k,\ S \cap R \neq \emptyset\}$for some$R \subset X$with$|R|=r$. This is best possible up to the size of the$o(1)$term, and improves a 1987 result of Frankl and F\"uredi, who obtained the same conclusion under the stronger hypothesis$k < (3-\sqrt{5})n/2$, in the case$r=2$. Our proof utilises an isoperimetric, influence-based method recently developed by Keller and the authors.Comment: 13 pages. Updated references, expositional changes and minor corrections following the helpful comments of an anonymous refere

A Hilton-Milner-type theorem and an intersection conjecture for signed sets

Abstract A family A of sets is said to be intersecting if any two sets in A intersect (i.e. have at least one common element). A is said to be centred if there is an element common to all the sets in A; otherwise, A is said to be non-centred. For any r ∈ [n] := {1, ..., n} and any integer k ≥ 2, let S n,r,k be the family . Let m := max{0, 2r − n}. We establish the following HiltonMilner-type theorems, the second of which is proved using the first: (i) If A 1 and A 2 are non-empty cross-intersecting (i.e. any set in A 1 intersects any set in A 2 ) sub-families of S n,r,k , then (ii) If A is a non-centred intersecting sub-family of S n,r,k , 2 ≤ r ≤ n, then We also determine the extremal structures. (ii) is a stability theorem that extends Erdős-Ko-Rado-type results proved by various authors. We then show that (ii) leads to further evidence for an intersection conjecture suggested by the author about general signed set systems