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Let the random variable $X, :=, e(mathcal{H}[B])$ count the number of edges of a hypergraph $mathcal{H}$ induced by a random $m$ element subset $B$ of its vertex set. Focussing on the case that $mathcal{H}$ satisfies some regularity condition we prove bounds on the probability that $X$ is far from its mean. It is possible to apply these results to discrete structures such as the set of $k$-term arithmetic progressions in the cyclic group $mathbb{Z}_N$. Furthermore, we show that our main theorem is essentially best possible and we deduce results for the case $Bsim B_p$ is generated by including each vertex independently with probability $p$.
Celebrated theorems of Roth and of Matouv{s}ek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $Theta(n^{1/4})$. We study the analogous problem in the $mathbb{Z}_n$ setting. We asymptoti
In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers $n$ and $k$, the expression $aw([n],k)$ denotes the smallest number of colors with which the integers ${
In this note we are interested in the problem of whether or not every increasing sequence of positive integers $x_1x_2x_3...$ with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms $x_i$, $x_j$, and $x_k$ such that $
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations.
We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerdens proof for the existence of arbitrary long monochromatic arithmet