26 problems

Let $R(g,n)$ denote the maximum size of a $B^{\ast}[g]$ set contained in $\{1,\dots,n\}$. A sequence $(S_n)_{n=1}^\infty$ of sets of positive integers becomes uniformly distributed…

For a positive integer $k$, let $S_k=\{n^k:n\in\mathbb{Z}_{>0}\}$; a set is Sidon if there are no nontrivial equal sums of two of its elements. Lander–Parkin–Selfridge's special-ca…

A finite Sidon set is a finite set of integers whose differences of distinct elements are all distinct. A finite perfect difference set is a finite set whose differences represent…

Let $S\subseteq \mathbb{F}_2^n$ be a $k$-cover, and suppose that … An AB function is a function $F\colon\mathbb{F}_2^{n/2}\to\mathbb{F}_2^{n/2}$ whose graph is denoted by…

Axis-value conjecture. For every $a\in\mathbb{F}_2^n$ and every nonzero $b\in\mathbb{F}_2^n$,

Uniform-exclude-distribution conjecture. If $d_{\mathcal{G}_F}$ is uniform on $\mathcal{Q}(\mathbb{F}_2^n,F)$, then $\mathcal{G}_F$ is maximal.

Single-point modification conjecture. The function $G$ is not APN.

Lazebnik–Verstraëte conjecture. For every integer $k\geq 3$, there exists a constant $c_k>0$ such that, for every $n\geq 1$, there exists a $k$-fold Sidon set $A$ in $\mathbb{Z}_n$…

Let $r\ge 2$ be an integer and let $n$ be sufficiently large. An $r$-coloring of a subset of $[n]$ is rainbow $[1,-1,1,-1]$-free if it contains no rainbow $4$-element set satisfyin…

Sidon-polynomial classification conjecture. If $P\in\mathbb{F}_p[x]$ is a Sidon polynomial, then either

Tait's perfect-difference-set extension conjecture. The set $S$ is contained in some perfect difference set $S'$ of size $p+1$.

Elementary subgroup conjecture. The set $T$ is a subgroup.

Subgroup-union conjecture. The set $T$ is the union of $O(1)$ subgroups.

Projective-plane completion conjecture. The action is faithful, $|\mathcal{P}|=(1+o(1))n$, and for some point $p$ and line $\ell$,

Klurman–Pohoata conjecture. There exists a constant $c>0$ such that, for every finite $A\subset\mathbb R$,

Let $k\geq 1$, $h\geq 2$, and $\varepsilon>0$. A $B_h[g]$ set $A\subseteq(\mathbb{Z}^{+})^k$ is a set of $k$-th powers with counting function $A(x)$. Generalized Sidon-set conjectu…

Let $A$ be a Sidon set in an ambient abelian group, and let $X$ and $Y$ be independent random variables taking values in $A$. Let $H$ denote Shannon entropy. Ajjanagadde's conjectu…

An $\alpha$-generalized Sidon set in $[n]=\{1,\ldots,n\}$ is a set containing at most $\alpha$ Sidon 4-tuples. Let $N_n(\alpha)$ denote the number of $\alpha$-generalized Sidon set…

Weak Anti-Freiman conjecture. Then $A$ contains either a $B_2[g]$ set or a $B^\circ_2[g]$ set of size at least $\delta'|A|$, where $g$ and $\delta'$ depend only on $\delta$.

An old problem of mine states as follows: Let $a_1 < a_2 < \cdots < a_k$ be a Sidon sequence. Can one extend it to a larger Sidon sequence … In other words, loosely speaking: can o…

More generally: Let $a_1 < a_2 < \cdots < a_{k_1} \leq n$, $b_1 < b_2 < \cdots < b_{k_2} \leq n$ be two Sidon sequences for which $a_j - a_i \neq b_v - b_u$ for all $i < j$ and…

Let $1 \leq a_1 < a_2 < \cdots < a_h \leq n$ be a maximum Sidon sequence. Can one find a Sidon sequence $b_1 < b_2 < \cdots < b_r \leq n$ for every $r$ and $n > n_0(r)$ so that the…

Here is an old conjecture of mine: Let $a_1 < a_2 < \ldots$ be an infinite sequence for which all the triple sums $a_i + a_j + a_k$ are distinct. Is it then true that…

Sidon also asked: Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence for which all the sums $a_i + a_j$ are distinct. Put … Probably there is a Sidon sequence $A$ for which ……

Sidon also asked: Let $1 \leq a_1 < a_2 < \cdots < a_k \leq n$ and assume that $a_i + a_j$ are all distinct. Put $h(n) = \max k$. Determine or estimate $h(n)$ as accurately as poss…