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We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1]d in $\mathbb{R}^d$ is (1 ± o(1)) $\frac{(2d-2)!}{(d-1)!}$n lnd−1n, if d is fixed. This estimate is relevant to analysis of the performance of exact algorithms for computing the largest empty axis-parallel box amidst n given points in an axis-parallel box R, especially the algorithms that proceed by examining all maximal empty boxes. Our method for bounding the expected number of maximal empty boxes also shows that the expected number of maximal empty orthants determined by n random points in $\mathbb{R}^d$ is (1 ± o(1)) lnd−1n, if d is fixed. This estimate is related to the expected number of maximal (or minimal) points amidst random points, and has application to algorithms for coloured orthogonal range counting.
We study Maker/Breaker games on the edges of sparse graphs. Maker and Breaker take turns at claiming previously unclaimed edges of a given graph H. Maker aims to occupy a given target graph G and Breaker tries to prevent Maker from achieving his goal. We show that for every d there is a constant c = c(d) with the property that for every graph G on n vertices of maximum degree d there is a graph H on at most cn edges such that Maker has a strategy to occupy a copy of G in the game on H.
This is a result about a game-theoretic variant of the size Ramsey number. For a given graph G, $\hat{r}'(G)$ is defined as the smallest number M for which there exists a graph H with M edges such that Maker has a strategy to occupy a copy of G in the game on H. In this language, our result yields that for every connected graph G of constant maximum degree, $\hat{r}'(G) = \Theta(n)$.
Moreover, we can also use our method to settle the corresponding extremal number for universal graphs: for a constant d and for the class ${\cal G}_{n}$ of n-vertex graphs of maximum degree d, $s({\cal G}_{n})$ denotes the minimum number such that there exists a graph H with M edges where, for everyG ∈ ${\cal G}_{n}$, Maker has a strategy to build a copy of G in the game on H. We obtain that $s({\cal G}_{n}) = \Theta(n^{2 - \frac{2}{d}})$.
Every graphon defines a random graph on any given number n of vertices. It was known that the graphon is random-free if and only if the entropy of this random graph is subquadratic. We prove that for random-free graphons, this entropy can grow as fast as any subquadratic function. However, if the graphon belongs to the closure of a random-free hereditary graph property, then the entropy is O(n log n). We also give a simple construction of a non-step-function random-free graphon for which this entropy is linear, refuting a conjecture of Janson.
We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice $\mathbb{Z}^d$, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion–exclusion.
This work studies the typical behaviour of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations. For instance, we show that a uniformly chosen M-Lipschitz function takes only M+1 values on most of the graph, with a double exponential decay for the probability of taking other values.
Given a set $A\subset\mathbb{Z}_{N}$, we say that a function $f\colon A \to \mathbb{Z}_{N}$ is a Freiman homomorphism if f(a)+f(b)=f(c)+f(d) whenever a,b,c,d ∈ A satisfy a+b=c+d. This notion was introduced by Freiman in the 1970s, and plays an important role in the field of additive combinatorics. We say that A is linear if the only Freiman homomorphisms are functions of the form f(x) = ax+b.
Suppose the elements of A are chosen independently at random, each with probability p. We shall look at the following question: For which values of p=p(N) is A linear with high probability as N → ∞? We show that if p=(2logN − ω(N))1/3N−2/3, where ω(N) → ∞ as N → ∞, then A is not linear with high probability, whereas if p=N−1/2+ε for any ε>0 then A is linear with high probability.
We generalize Reimer's Inequality [6] (a.k.a. the BKR Inequality or the van den Berg–Kesten Conjecture [1]) to the setting of finite distributive lattices.
Let m,n and t be positive integers. Consider [m]n as the set of sequences of length n on an m-letter alphabet. We say that two subsets A⊂[m]n and B⊂[m]n cross t-intersect if any two sequences a∈A and b∈B match in at least t positions. In this case it is shown that if $m > (1-\frac 1{\sqrt[t]2})^{-1}$ then |A||B|≤(mn−t)2. We derive this result from a weighted version of the Erdős–Ko–Rado theorem concerning cross t-intersecting families of subsets, and we also include the corresponding stability statement. One of our main tools is the eigenvalue method for intersection matrices due to Friedgut [10].