269 problems

Let $M_3(n)$ denote the maximum size of a family $\mathcal{F}\subseteq\binom{[n]}{4}$ with $\operatorname{VC}(\mathcal{F})\le 3$. The paper establishes the lower bound … for every…

Manickam–Miklós–Singhi conjecture. There exist at least $\binom{n-1}{k-1}$ subsets $A\subseteq [n]$ such that $|A|=k$ and

For $d \ge 2$, a $d$-cluster is a collection of $d$ subsets of size $k$ of $[n]$ with trivial intersection and union of size at most $2k$; a family containing no such collection is…

Johnson–Leader–Russell conjecture. Extremal families—families of a given size having the maximum possible number of maximal chains—arise from two-level posets, closely related to t…

Let $X$ be an $n$-element ground set. Two subsets $A,B\subset X$ are crossing if none of the four sets $A\setminus B$, $B\setminus A$, $A\cap B$, and $X\setminus(A\cup B)$ are empt…

Gerbner et al.'s conjecture. If $n$ is sufficiently large compared to $k$, then the minimum size of a saturated $k$-Sperner system is

Leclerc–Zelevinsky's purity conjecture. The set $\mathbf W_n$ is pure.

Erdős matching conjecture. For all $n\geq (s+1)k-1$, if $\nu(\mathcal{H})\leq s$, then

Let $[n]=\{1,\ldots,n\}$, and let $h(n,k,d,s)$ denote the maximum size of a family of $k$-subsets of $[n]$ that contains no $d$ sets with union of size at most $s$ and empty inters…

Let $F$ be a $k\times \ell$ matrix with $F\ne \begin{bmatrix}0\\ 1\end{bmatrix}$. Let $I_k$ be the $k\times k$ identity matrix, $I_k^c$ its (0,1)-complement, and $T_k$ the…

Let $n=s(m+1)-\ell$ and define … For $s\ge2$, $m\ge1$, and $1\le\ell\le\lceil s/2\rceil$, this family is a weighted construction with matching number less than $s$. Frankl–Kupavski…

For integers $N,k,t$ with $Nge kt$, let $e_k(N,t)$ be the maximum size of a $k$-uniform family in $\binom{[N]}k$ with matching number less than $t$. Erdős's matching conjecture. ……

Erdős–Kleitman conjecture. If $\mathcal{F}$ contains no matching of size $s$ and is maximal with respect to this property, then

Hegedüs' conjecture. Every such system satisfies

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{F}\subseteq\binom{[n]}{d+1}$ be a $(d+1)$-uniform set family. Its VC-dimension is the largest size of a set $S\subseteq[n]$ such that ev…

Let ${\cal F}$ be a family of sets. Call ${\cal F}$ hereditary if every subset of every member of ${\cal F}$ also belongs to ${\cal F}$. Call ${\cal F}$ EKR if some element $x$ sat…

Let $\mathcal{F}$ be a union-closed family of sets. A power set is a family of the form $\mathcal{P}(X)$ for some set $X$. Poonen's conjecture. Unless $\mathcal{F}$ is a power set,…

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…

Erdős's matching conjecture. If $\mathcal{F}\subset\binom{[n]}{k}$ satisfies $\nu(\mathcal{F})\leqslant s$, then

Let $m(n,n,1)$ be the maximum size of a $1$-cross intersecting set pair system in which $|A_i|\leq n$ and $|B_i|\leq n$ for every $i$. The classical Bollobás bound gives…

Huang's conjecture. If $\mathcal{F}\subset 2^{[n]}$ is intersecting, then

Bollobás–Leader conjecture. For every $s$, there is some $1\leq \ell\leq k$ such that an $ℓ$-ball minimizes the number of disjoint pairs among all $k$-uniform families of size $s$.

Let $n$ be a positive integer and let $l$ satisfy … for some fixed $\epsilon>0$. An $l$-avoiding family is a family ${\mathcal A}\subseteq{\mathcal P}[n]$ containing no two sets wh…

A Sperner $k$-partition system on an $n$-set is a family of $k$-partitions such that no class of one partition is contained in a class of another. A partition is almost uniform if…

Larman's conjecture. The family $K$ can be partitioned into $n$ parts so that in each part every two members have $(t+1)$ elements in common.