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

Vertex Isoperimetry and Independent Set Stability for Tensor Powers of Cliques

The tensor power of the clique on t vertices (denoted by K_t^n) is the graph on vertex set {1, ..., t}^n such that two vertices x, y in {1, ..., t}^n are connected if and only if x_i != y_i for all i in {1, ..., n}. Let the density of a subset S of K_t^n to be mu(S) := |S|/t^n. Also let the vertex boundary of a set S to be the vertices of the graph, including those of S, which are incident to some vertex of S. We investigate two similar problems on such graphs. First, we study the vertex isoperimetry problem. Given a density nu in [0, 1] what is the smallest possible density of the vertex boundary of a subset of K_t^n of density nu? Let Phi_t(nu) be the infimum of these minimum densities as n -> infinity. We find a recursive relation allows one to compute Phi_t(nu) in time polynomial to the number of desired bits of precision. Second, we study given an independent set I of K_t^n of density mu(I) = (1-epsilon)/t, how close it is to a maximum-sized independent set J of density 1/t. We show that this deviation (measured by mu(IJ)) is at most 4 epsilon^{(log t)/(log t - log(t-1))} as long as epsilon < 1 - 3/t + 2/t^2. This substantially improves on results of Alon, Dinur, Friedgut, and Sudakov (2004) and Ghandehari and Hatami (2008) which had an O(epsilon) upper bound. We also show the exponent (log t)/(log t - log(t-1)) is optimal assuming n tending to infinity and epsilon tending to 0. The methods have similarity to recent work by Ellis, Keller, and Lifshitz (2016) in the context of Kneser graphs and other settings. The author hopes that these results have potential applications in hardness of approximation, particularly in approximate graph coloring and independent set problems

On a biased edge isoperimetric inequality for the discrete cube

The `full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary $\partial A$ of a set $A \subset \{0,1\}^n$, as a function of $|A|$. A weaker (but more widely-used) lower bound is $|\partial A| \geq |A| \log_2(2^n/|A|)$, where equality holds iff $A$ is a subcube. In 2011, the first author obtained a sharp `stability' version of the latter result, proving that if $|\partial A| \leq |A| (\log(2^n/|A|)+\epsilon)$, then there exists a subcube $C$ such that $|A \Delta C|/|A| = O(\epsilon /\log(1/\epsilon))$. The `weak' version of the edge isoperimetric inequality has the following well-known generalization for the `$p$-biased' measure $\mu_p$ on the discrete cube: if $p \leq 1/2$, or if $0 < p < 1$ and $A$ is monotone increasing, then $p\mu_p(\partial A) \geq \mu_p(A) \log_p(\mu_p(A))$. In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if $p\mu_p(\partial A) \leq \mu_p(A) (\log_p(\mu_p(A))+\epsilon)$, then there exists a subcube $C$ such that $\mu_p(A \Delta C)/\mu_p(A) = O(\epsilon' /\log(1/\epsilon'))$, where $\epsilon' =\epsilon \ln (1/p)$. This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd\H{o}s-Ko-Rado type theorems in extremal combinatorics, including the seminal `complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the `full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the `full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem.Comment: 36 pages. More explanations added, and minor corrections made, in response to referee comment

Topics in graph colouring

In this thesis, we study two variants of graph (vertex) colourings: multicolouring and correspondence colouring. In ordinary graph colouring, each vertex receives a colour. Such a colouring is proper if adjacent vertices receive different colours. In k-multi-colouring, each vertex receives a set of k colours, and such a multi-colouring is proper if adjacent vertices receive disjoint set of colours. A graph is (n, k)-colourable if there is a proper k-multi-colouring of it using n colours. In the first part of the thesis, we study the following two questions. 1. For given n, k and n′, k′, if a graph is (n, k)-colourable, then what is the largest subgraph of it that is (n′, k′)-colourable? 2. For given n, k, if a graph is (n, k)-colourable, then for what n′, k′ is the whole graph (n′, k′)-colourable? Question 1 is inspired by a partial colouring conjecture asked by Albertson, Grossman, and Haas [2] in 2000 regarding list colouring. We obtain exact answers for specific values of the parameters, and upper and lower bounds on the largest (n′, k′)-colourable subgraph for general values of n′, k′. For Question 2, we first observe how it can be reformulated into a conjecture by Stahl from 1976 regarding Kneser graphs, and prove new results towards Stahl’s conjecture. In the second part of the thesis, we study another variant of colouring, which is known as correspondence colouring. In correspondence colouring, each vertex is associated with a prespecified list of colours, and there is prespecified correspondence associated with each edge specifying which pair of colours from the two endvertices correspond. (On each edge, a colour on one endvertex corresponds to at most one colour on the other endvertex.) A correspondence colouring is proper if each vertex receives a colour from its prespecified list, and that for each edge, the colours on its endvertices do not correspond. A graph is n-correspondence-colourable if a proper correspondence colouring exist for any prespecified correspondences on any prespecified n-colour-lists associated to each vertex. As correspondence colouring is a generalisation of list colouring, it is natural to ask whether Albertson, Grossman, and Haas’ conjecture can be generalised to correspondence colouring. Unfortunately, there are graphs on which their conjectured value does not hold, and we will present a series of them. We then study: for given n and n′, how many vertices of a n-correspondence colourable graph can always be properly correspondence-coloured with arbitrary correspondences and arbitrary n′-colour-lists on that graph? We generalise some results from the original conjecture in list colouring. Then we discuss some sufficient conditions for a proper correspondence colouring to exist. The correspondence chromatic number of a graph is the smallest n such that the graph is n-correspondence colourable. We study how different graph operations affect the correspondence chromatic number of multigraphs, in which multiple edges are allowed