On the Union of Arithmetic Progressions

We show that for every $\varepsilon>0$ there is an absolute constant $c(\varepsilon)>0$ such that the following is true. The union of any $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences must consist of at least $c(\varepsilon)n^{2-\varepsilon}$ elements. We observe, by construction, that one can find $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences such that the cardinality of their union is $o(n^2)$. We refer also to the non-symmetric case of $n$ arithmetic progressions, each of length $\ell$, for various regimes of $n$ and $\ell$

Anti-Ramsey numbers of small graphs

The anti-Ramsey number $AR(n,G$), for a graph $G$ and an integer $n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy of $G$, namely, a copy of $G$ whose edges have distinct colours. In this paper we determine the anti-Ramsey numbers of all graphs having at most four edges

Semi-random process without replacement

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some pre-determined objective in an online randomized environment. We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced in \cite{BHKPSS}. The process starts with an empty graph on the vertex set $[n]$. For every positive integer $k$, in the $k$th round of the process, the decision-maker, called \emph{Builder}, is offered the vertex $v_k := \pi_{\lceil k/n \rceil}(k - \lfloor (k-1)/n \rfloor n)$, where $\pi_1, \pi_2, \ldots$ is a sequence of permutations in $S_n$, chosen independently and uniformly at random. Builder then chooses an additional vertex (according to a strategy of his choice) and connects it by an edge to $v_k$. For several natural graph properties, such as $k$-connectivity, minimum degree at least $k$, and building a given spanning graph (labeled or unlabeled), we determine the typical number of rounds Builder needs in order to construct a graph having the desired property. Along the way we introduce and analyze two urn models which may also have independent interest

The maximal number of 33-term arithmetic progressions in finite sets in different geometries

Green and Sisask showed that the maximal number of $3$-term arithmetic progressions in $n$-element sets of integers is $\lceil n^2/2\rceil$; it is easy to see that the same holds if the set of integers is replaced by the real line or by any Euclidean space. We study this problem in general metric spaces, where a triple $(a,b,c)$ of points in a metric space is considered a $3$-term arithmetic progression if $d(a,b)=d(b,c)=\frac{1}{2}d(a,c)$. In particular, we show that the result of Green and Sisask extends to any Cartan--Hadamard manifold (in particular, to the hyperbolic spaces), but does not hold in spherical geometry or in the $r$-regular tree, for any $r\geq 3$

Non-constant ground configurations in the disordered ferromagnet

The disordered ferromagnet is a disordered version of the ferromagnetic Ising model in which the coupling constants are non-negative quenched random. A ground configuration is an infinite-volume configuration whose energy cannot be reduced by finite modifications. It is a long-standing challenge to ascertain whether the disordered ferromagnet on the $\mathbb{Z}^D$ lattice admits non-constant ground configurations. We answer this affirmatively in dimensions $D\ge 4$, when the coupling constants are sampled independently from a sufficiently concentrated distribution. The obtained ground configurations are further shown to be translation-covariant with respect to $\mathbb{Z}^{D-1}$ translations of the disorder. Our result is proved by showing that the finite-volume interface formed by Dobrushin boundary conditions is localized, and converges to an infinite-volume interface. This may be expressed in purely combinatorial terms, as a result on the fluctuations of certain minimal cutsets in the lattice $\mathbb{Z}^D$ endowed with independent edge capacities.Comment: A combinatorial interpretation was added; some additional minor changes to the presentatio