In this paper we give a short proof of the Random Ramsey Theorem of Rödl and Ruciński: for any graph F which contains a cycle and r ≥ 2, there exist constants c, C > 0 such that

$$\begin{equation*}\Pr[G_{n,p} \rightarrow (F)_r^e] =\begin{cases}1-o(1) &p\ge Cn^{-1/m_2(F)},\\o(1) &p\le cn^{-1/m_2(F)},\end{cases}\end{equation*}$$

$$\begin{equation*}m_2(F) = \max_{J\subseteq F, v_J\ge 2} \frac{e_J-1}{v_J-2}.\end{equation*}$$

We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least

\begin{equation*}\binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}.\end{equation*}

Let ${\mathrm{OT} }_{d} (n)$ be the smallest integer $N$ such that every $N$-element point sequence in ${ \mathbb{R} }^{d} $ in general position contains an order-type homogeneous subset of size $n$, where a set is order-type homogeneous if all $(d+ 1)$-tuples from this set have the same orientation. It is known that a point sequence in ${ \mathbb{R} }^{d} $ that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in ${ \mathbb{R} }^{d} $. Two famous theorems of Erdős and Szekeres from 1935 imply that ${\mathrm{OT} }_{1} (n)= \Theta ({n}^{2} )$ and ${\mathrm{OT} }_{2} (n)= {2}^{\Theta (n)} $. For $d\geq 3$, we give new bounds for ${\mathrm{OT} }_{d} (n)$. In particular, we show that ${\mathrm{OT} }_{3} (n)= {2}^{{2}^{\Theta (n)} } $, answering a question of Eliáš and Matoušek, and, for $d\geq 4$, we show that ${\mathrm{OT} }_{d} (n)$ is bounded above by an exponential tower of height $d$ with $O(n)$ in the topmost exponent.

We generalise an argument of Leader, Russell, and Walters to show that almost all sets of $d+ 2$ points on the $(d- 1)$-sphere ${S}^{d- 1} $ are not contained in a transitive set in some ${\mathbf{R} }^{n} $.

Let β>1 be a real number, and let {ak} be an unbounded sequence of positive integers such that ak+1/ak≤β for all k≥1. The following result is proved: if n is an integer with n>(1+1/(2β))a1 and A is a subset of {0,1,…,n} with , then (A+A)∩(A−A)contains a term of {ak }. The lower bound for |A| is optimal. Beyond these, we also prove that if n≥3is an integer and A is a subset of {0,1,…,n} with , then (A+A)∩(A−A)contains a power of 2. Furthermore, cannot be improved.

Given a commutative semigroup (S, +) with identity 0 and u × v matrices A and B with nonnegative integers as entries, we show that if C = A – B satisfies Rado's columns condition over ℤ, then any central set in S contains solutions to the system of equations . In particular, the system of equations is then partition regular. Restricting our attention to the multiplicative semigroup of positive integers (so that coefficients become exponents) we show that the columns condition over ℤ is also necessary for the existence of solutions in any central set (while the distinct notion of the columns condition over Q is necessary and sufficient for partition regularity over ℕ\{1}).