Related papers: Is it true that most sets are Sidon?

For a fixed poset $\mathcal P$ we say that a family $\mathcal F\subseteq\mathcal P([n])$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add a new set to $\mathcal F$, we form an induced…

Strong external difference families (SEDFs) are much-studied combinatorial objects motivated by an information security application. A well-known conjecture states that only one abelian SEDF with more than 2 sets exists. We show that if the…

Let $\varphi (x_{1}, \ldots, x_{h})=c_{1} x_{1}+\cdots+c_{h} x_{h}$ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $\varphi$-Sidon set if…

We prove that Bourbaki's mathematics is incomplete.

Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers can be written as the projection of a spectrahedron. Recently, Scheiderer disproved this conjecture. We show, however, that the following result,…

Let $\mathcal{F}$ be any collection of linearly separable sets of a set $P$ of $n$ points either in $\mathbb{R}^2$, or in $\mathbb{R}^3$. We show that for every natural number $k$ either one can find $k$ pairwise disjoint sets in…

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers,…

We present a new method to obtain infinite Sidon sequences, based on the discrete logarithm. We construct an infinite Sidon sequence A, with A(x)= x^{\sqrt 2-1+o(1)}. Ruzsa proved the existence of a Sidon sequence with similar counting…

Let $h \geq 2$ and let ${ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of sets of integers. For nonzero integers $c_1,\ldots, c_h$, consider the linear form $\varphi = c_1 x_1 + c_2x_2 + \cdots + c_h x_h$. The \emph{representation…

The problem of determining the Tur\'an number of $C_4$ is a well studied problem that dates back to a paper of Erd\"os from 1938. It is known that Sidon sets can be used to construct $C_4$-free graphs. If $\A$ is a Sidon set in the abelian…

Classical strong external difference families (SEDFs) are much-studied combinatorial structures motivated by information security applications; it is conjectured that only one classical abelian SEDF exists with more than two sets. Recently,…

A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric…

Let $f$ be a nonnegative function supported on $(-1/4, 1/4)$. We show $$ \sup_{x \in \mathbb{R}}{\int_{\mathbb{R}}{f(t)f(x-t)dt}} \geq 1.28\left(\int_{-1/4}^{1/4}{f(x)dx} \right)^2,$$ where 1.28 improves on a series of earlier results. The…

Let $p$ be a prime, and $q=p^n$ be a prime power. In his works on Sidon sets over $\mathbb{F}_q\times \mathbb{F}_q$, Cilleruelo conjectured about polynomials that could generate $q$-element Sidon sets over $\mathbb{F}_q\times \mathbb{F}_q$.…

A subspace of a finite field is called a Sidon space if the product of any two of its nonzero elements is unique up to a scalar multiplier from the base field. Sidon spaces, introduced by Roth et al. (IEEE Trans Inf Theory 64(6): 4412-4422,…

We summarize some facts on chains (totally ordered sets), from an order-theoretic and from a topological point of view. We highlight the fact that many classical theorems that are true for partially ordered sets under some completeness…

A subspace of a finite extension field is called a Sidon space if the product of any two of its elements is unique up to a scalar multiplier from the base field. Sidon spaces were recently introduced by Bachoc et al. as a means to…

In this paper we show that for every $2\leq n\in \mathbb{N}$, the statement "there is an $n$-entangled set, but there are no $n+1$-entangled sets" is consistent. We also prove some theorems which improve our understanding of entangled sets…

Erd\H{o}s Problem 30 asks for sharp asymptotics of the Sidon extremal function $h(N)$, and Singer's construction is the classical source of lower-bound examples matching the main term. We present a Lean 4 formalization of Singer's Sidon set…

In the paper we present a proof of the local criterion for crystalline structures which generalizes the local criterion for regular systems. A Delone set is called a crystal if it is invariant with respect to a crystallgraphic group.…