4 problems

Let $f(n)$ be the largest cardinality of a Sidon set contained in $\{1,2,\ldots,n\}$, where a subset $\mathcal{S}$ of the natural numbers is Sidon if all sums $s_1+s_2$ with…

For $n\in\mathbb{N}$, let $R_{2}(n;\mathfrak{X}_{1})$ denote the number of representations of $n$ as a sum of two elements of $\mathfrak{X}_{1}\subset\mathbb{N}_{0}$. Erdős's conje…

Let $h\geqslant 2$ and let $f$ be a function for which the notion of a $\Theta(f)$-regular asymptotic $h$-basis is defined. Adapted Erdős 1956 conjecture. There are no $\Theta(f)$-…

Let $h\geqslant 2$, let $\mathcal{S}$ be the space of subsets of the nonnegative integers, and for an asymptotic $h$-basis $A$ let $r_h(n;A)$ denote its representation function. Er…