163 problems

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.

Let $n=4k+7$ with $k\geq 2$, and let $\operatorname{Cone}=\operatorname{Cone}(\operatorname{Cone}_{2k+1}\cup\operatorname{Cone}_{2k+3})$. Cone asymptotic-size conjecture. Is … Nume…

Let $n=4k+7$ and let $\mathcal{D}$ be the constructed maximal commutative algebraic system in $\mathcal{P}_n(*)$. $\mathcal{D}$ asymptotic-size conjecture. Is … Numerical checks th…

Let $k>d\ge 2$ and let $n\ge k(d+1)/d$. A $d$-simplex-cluster is a collection of $d+1$ distinct members $A_1,\dots,A_{d+1}$ of a family $\mathcal F\subseteq\binom{[n]}{k}$ such tha…

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 $\mathcal{S}\subseteq\mathcal{P}([n])$. Suppose that $\mathcal{S}$ satisfies Reimer's conditions: there is a filter…

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…

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…

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 $\mathcal{F}\subseteq 2^{[n]}$. 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 $F$ 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…

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…

KleItman's conjecture. Let $n,M>0$ and $k\geq 2$ be integers. Among families $\mathcal{F}\subseteq\mathcal{P}(n)$ of size $M$, the number of $k$-chains in $\mathcal{F}$ is minimize…

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…

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

Frankl–Wang conjecture. Under these hypotheses,

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.