We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing
property for Voronoi percolation at criticality, and an Efron-Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.
Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the two followi
ng cases: we say $T$ is explosive if [ lim_{nto infty}m_n(T) < infty, ] and say that $T$ exhibits linear growth if [ liminf_{nto infty} frac{m_n(T)}{n} > 0. ] We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres. As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.
The Ramsey number r(K_s,Q_n) is the smallest positive integer N such that every red-blue colouring of the edges of the complete graph K_N on N vertices contains either a red n-dimensional hypercube, or a blue clique on s vertices. Answering a questio
n of Burr and ErdH{o}s from 1983, and improving on recent results of Conlon, Fox, Lee and Sudakov, and of the current authors, we show that r(K_s,Q_n) = (s-1) (2^n - 1) + 1 for every s in N and every sufficiently large n in N.
The areas of Ramsey theory and random graphs have been closely linked ever since ErdH{o}s famous proof in 1947 that the diagonal Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was introduced as a model
which might potentially provide good lower bounds for the off-diagonal Ramsey numbers $R(3,k)$. In this model, edges of $K_n$ are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted $G_{n,triangle}$. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kims celebrated result that $R(3,k) = Theta big( k^2 / log k big)$. In this paper we improve the results of both Bohman and Kim, and follow the triangle-free process all the way to its asymptotic end. In particular, we shall prove that $$ebig( G_{n,triangle} big) ,=, left( frac{1}{2sqrt{2}} + o(1) right) n^{3/2} sqrt{log n },$$ with high probability as $n to infty$. We also obtain several pseudorandom properties of $G_{n,triangle}$, and use them to bound its independence number, which gives as an immediate corollary $$R(3,k) , ge , left( frac{1}{4} - o(1) right) frac{k^2}{log k}.$$ This significantly improves Kims lower bound, and is within a factor of $4 + o(1)$ of the best known upper bound, proved by Shearer over 25 years ago.
The Ramsey number r(K_3,Q_n) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K_N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and ErdH{o}s conjectured
that r(K_3,Q_n) = 2^{n+1} - 1 for every n in N, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K_3,Q_n) le 7000 cdot 2^n. Here we show that r(K_3,Q_n) = (1 + o(1)) 2^{n+1} as n to infty.
We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c e 1/2. Our proof
uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p_c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.
We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson in the case of unoriented
graphs, and by Chung and Graham in the case of tournaments. Indeed, our main theorem extends to the case of a general underlying graph G the main result of Chung and Graham which corresponds to the case that G is complete. One interesting aspect of these results is that exactly two of the four orientations of a four-cycle can be used for a quasi-randomness condition, i.e., if the number of appearances they make in D is close to the expected number in a random orientation of the same underlying graph, then the same is true for every small oriented graph H
This paper has been withdrawn by the author.
A $k$-sum of a set $Asubseteq mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $kwedge A$ for the set of $k$-sums of $
A$ we prove that [ frac{|(k+1)wedge A|}{|kwedge A|}, le , frac{|A|-k}{k+1} ] whenever $|A|ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.
It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an active are
a of research, and is relatively well-understood in the theory of quasi-random graphs and extremal combinatorics. In this paper, we consider the simplest cases of such a general question for oriented graphs, and provide some results on the global behavior of the orientation of $D$. For the case that $H$ is an oriented four-cycle we prove: in every $H$-free oriented graph $D$, there is a pair $A,Bssq V(D)$ such that $e(A,B)ge e(D)^{2}/32|D|^{2}$ and $e(B,A)le e(A,B)/2$. We give a random construction which shows that this bound on $e(A,B)$ is best possible (up to the constant). In addition, we prove a similar result for the case $H$ is an oriented six-cycle, and a more precise result in the case $D$ is dense and $H$ is arbitrary. We also consider the related extremal question in which no condition is put on the oriented graph $D$, and provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures.
