162 problems

For integers $k\ge 2$, $s\ge 1$, and $n\ge (s+1)k$, every family $\mathcal{F}\subseteq\binom{[n]}{k}$ with matching number $\nu(\mathcal{F})\le s$ satisfies …

For every integer $n\ge 1$ and every integer $s$ satisfying $1\le s\le 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}$, let $\mathcal F$ be a family of even-sized subsets of $[n]$ w…

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{F}\subseteq 2^{[n]}$. The family $\mathcal{F}$ is $t$-intersecting if $|A\cap B|\geq t$ for all $A,B\in\mathcal{F}$, and it is $k$-Spern…

Let $[n]=\{1,\ldots,n\}$ and let $2^{[n]}$ denote its power set. For a family $\mathcal{F}\subset 2^{[n]}$, write … and define its diametral overflow by … where…

Conjecture for totally chain-intersecting families. If $\mathcal F$ is a totally $(p,q)$-chain intersecting family and $p\geq q$, then

For integers $k\geq 2$, let $n=4k+7$ and $m=2k+3$. Let $\mathrm{Spec}(k,2)$ be the minimum cardinality of a $(2k+3)$-bicommutative system in $\mathcal{P}_{n}(m)$. Asymptotic bicomm…

Minimal-generator extension conjecture. If $\mathcal{B}$ is not $FC$, then $\mathcal{B}^{\prime}$ is not $FC$ either. This conjecture proposes that replacing a minimal generator by…

Almost-uniform extremal conjecture. For positive integers $n,k$, a largest Sperner partition system in $\mathcal{P}^n_k$ is an almost-uniform partition system.

Write $n=sm+c=s(m+1)-\ell$, and for $0\le k\le m$ define … and let $\mathcal P(m,s,\ell)$ be the shifted family $\{F\subseteq[n]\colon |F|+|F\cap[\ell-1]|\ge m+1\}$. Completeness c…

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…

Let $[n]=\{1,2,\ldots,n\}$, let $e(n,s)$ be the maximum of $|\mathcal F|$ over families $\mathcal F\subseteq 2^{[n]}$ with matching number $\nu(\mathcal F)<s$, and, for a weight fu…

Catalan classification conjecture. There are exactly $C_n$ families of size $2^n$ in which every two distinct paths have an even number of common edges.

Sharpening of Szabó's conjecture. Equality holds in this lower bound for every $N\ge 1$, that is,

Frankl–Wang conjecture. Under these hypotheses,

Let $[n]:=\{1,2,\ldots,n\}$. For $k\geq 3$, a family $\mathcal{F}$ of subsets of $[n]$ is $k$-sunflower free if it contains no $k$ pairwise distinct sets whose pairwise intersectio…

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…

Hegedüs's conjecture. If $\mathcal{F}\subseteq 2^{[n]}$ is a Sperner family and $|\mathcal{F}|\ge n+1$, then $\mathcal{F}$ is $2$-balanced.

Let $m_{k,n}(a,a)$ be the maximum size of a family of subsets of $[n]$ whose intersections are all congruent to $a$ modulo $k$, with $n\geq k\geq 2$. Parity conjecture. If…

Let $m_{k,n}(a,b)$ denote the maximum size of a family of subsets of $[n]$ whose pairwise intersections are congruent to $a$ modulo $k$ when the two subsets coincide and to $b$ mod…

Let $m_{k,n}(a,b)$ denote the maximum size of a family of subsets of $[n]$ whose pairwise intersections are congruent to $a$ modulo $k$ when the two subsets coincide and to $b$ mod…

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

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{F}\subseteq 2^{[n]}$ be a non-empty union-closed family, meaning that $A\cup B\in\mathcal{F}$ for all $A,B\in\mathcal{F}$. Let…

For integers $n\ge s\ge2$, define … For integers $m\ge1$ and $1\le\ell\le\lceil s/2\rceil$, set $n=(m+1)s-\ell$ and, for any $L\in\binom{[n]}{\ell-1}$, define … and write…

Let $\mathcal S \subseteq 2^{[m]}$ be a set system, let $w:[m]\rightarrow\mathbb{R}_{\geq 0}$ be weights, and let $\epsilon>0$. Write $\operatorname{CL}(\mathcal S)$ for the chain…

Let $X$ be an $n$-element set, and let $\binom{X}{\bullet}$ denote the Boolean lattice of subsets of $X$. For integers $t,k\geq 1$ with $n=tk$, partition $X$ into pairwise disjoint…