Lines in Euclidean Ramsey theory

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $\ell_m$ for any $m \geq 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number $n$, there is a set $K \subset \mathbb{E}^1$ and a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $K$.Comment: 7 page

Large subgraphs without complete bipartite graphs

In this note, we answer the following question of Foucaud, Krivelevich and Perarnau. What is the size of the largest $K_{r,s}$-free subgraph one can guarantee in every graph $G$ with $m$ edges? We also discuss the analogous problem for hypergraphs.Comment: 4 page

Exotic Decays of Heavy B quarks

Heavy vector-like quarks of charge $-1/3$, $B$, have been searched for at the LHC through the decays $B\rightarrow bZ,\, bh,\,tW$. In models where the $B$ quark also carries charge under a new gauge group, new decay channels may dominate. We focus on the case where the $B$ is charged under a $U(1)^\prime$ and describe simple models where the dominant decay mode is $B\rightarrow bZ^\prime\rightarrow b (b\bar{b})$. With the inclusion of dark matter such models can explain the excess of gamma rays from the Galactic center. We develop a search strategy for this decay chain and estimate that with integrated luminosity of 300 fb$^{-1}$ the LHC will have the potential to discover both the $B$ and the $Z'$ for $B$ quarks with mass below $\sim 1.6$ TeV, for a broad range of $Z'$ masses. A high-luminosity run can extend this reach to $2$ TeV.Comment: 28 pages, 10 figures, 3 table

The Green-Tao theorem: an exposition

The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.Comment: 26 pages, 4 figure

ENVIRONMENTAL POLICY CONSIDERATIONS IN THE GRAIN-LIVESTOCK SUBSECTORS IN CANADA, MEXICO AND THE UNITED STATES

Environmental Economics and Policy,

Extremal results in sparse pseudorandom graphs

Szemer\'edi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemer\'edi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.Comment: 70 pages, accepted for publication in Adv. Mat

Tower-type bounds for unavoidable patterns in words

A word $w$ is said to contain the pattern $P$ if there is a way to substitute a nonempty word for each letter in $P$ so that the resulting word is a subword of $w$. Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised the patterns $P$ which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains $P$. Zimin's characterisation says that a pattern is unavoidable if and only if it is contained in a Zimin word, where the Zimin words are defined by $Z_1 = x_1$ and $Z_n=Z_{n-1} x_n Z_{n-1}$. We study the quantitative aspects of this theorem, obtaining essentially tight tower-type bounds for the function $f(n,q)$, the least integer such that any word of length $f(n, q)$ over an alphabet of size $q$ contains $Z_n$. When $n = 3$, the first non-trivial case, we determine $f(n,q)$ up to a constant factor, showing that $f(3,q) = \Theta(2^q q!)$.Comment: 17 page