18 problems

For a sunflower-free $k$-uniform family, its base elements are the elements contained in at least one set of the family. Sunflower-free base-size exponential bound conjecture. The…

Extremal vector-space sunflower conjecture. There exists a constant $C=C(q,s)$ such that every $s$-sunflower-free family ${\mathcal F}$ of $k$-spaces over the field with $q$ elemen…

A $k$-uniform family is a family of sets in which every set has size $k$. An odd-sunflower is a family of at least two sets such that every element of the underlying set belongs to…

Slow-growth conjecture. Suppose $f:\mathbb{N}\to\mathbb{N}$ is non-decreasing and tends to infinity. There is a constant $c_f\in\mathbb{N}$ such that, whenever…

Let $\mathcal F$ be a family of sets, each of cardinality at most $m$, and let $k$ be a positive integer. A $k$-sunflower is a family of $k$ different sets with common pair-wise in…

Erdős–Rado conjecture. For fixed $k$, the upper bound can be improved to

For each integer $r\geq 3$, let $V\subset\mathbb R^3$ and let $\mathcal F$ be a family of subsets of $V$, each of size $k$, such that every member of $\mathcal F$ is the intersecti…

Let $f^d_r(k)$ be the least positive integer $m$ such that every family $\mathcal F$ of $k$-sets with $|\mathcal F|\geq m$ and $\text{VC-dim}(\mathcal F)\leq d$ contains an $r$-sun…

Let a near-sunflower of size $r$ be the configuration defined in the paper, and let $\mathcal{F}$ be a family of $k$-element sets. Near-sunflower exponential-bound conjecture. For…

Let $r\geq 1$ and let $X$ be a finite ground set. Let $\mathcal{F}$ be a $w$-set system, meaning that every member has size at most $w$. A random $\delta$-fraction coloring colors…

Let $X$ be a finite ground set and let $\mathcal{F}$ be a $w$-set system, meaning that every member of $\mathcal{F}$ has size at most $w$. Color each element of $X$ independently a…

Bounded subcover conjecture. There exist $F_1,\ldots,F_{2k}\in\cal F$ such that

Linear-union conjecture. There exists a constant $D>0$ such that

Let $k\leq D$, and let $A\subset(\mathbb Z/D\mathbb Z)^n$. A collection of vectors is a $k$-sunflower if, in every coordinate, its entries are either all different or all equal;…

For integers $n$ and $k$, let ${\cal F}(n,k)$ be the collection of $k$-tuples of families ${\cal A}_i\subset 2^{[n]}$ that contain no multicolor sunflower with $k$ petals, and defi…

Let ${\mathbb Z}_D=\{1,\ldots,D\}$, and call three vectors in ${\mathbb Z}_D^n$ a $3$-sunflower when their entries in each coordinate are either all equal or all distinct. Alon–Shp…

Let ${\mathbb Z}_D=\{1,\ldots,D\}$. Vectors $v_1,v_2,v_3\in{\mathbb Z}_D^n$ form a $3$-sunflower if, in every coordinate, their entries are either all equal or all distinct. Alon–S…

Let $X$ be a universal set, let ${\mathcal F}$ be a family of sets each of cardinality at most $s$, and let $k$ be a positive integer. A $k$-sunflower is a sunflower consisting of…