125 problems

Asymptotic product conjecture. For fixed $k\geq 2$ and sufficiently large $n$ with respect to $k$,

For each $k$, let $m_{\ell}(k)$ denote the extremal quantity corresponding to the $\ell$-diversity introduced in the paper. Asymptotic vanishing-ratio conjecture. For all…

Let $n,s$ be integers with $n\geq 2s-1$, let $L=\{\ell_{1},\ell_{2},\ldots,\ell_{s}\}$ be a set of $s$ non-negative integers, and let an $L$-intersecting Sperner system be a Sperne…

Let $L=\{\ell_{1},\ell_{2},\ldots,\ell_{s}\}$ be a set of $s$ non-negative integers and $K=\{k_{1},k_{2},\ldots,k_{r}\}$ be a set of $r$ positive integers with … Let an $L$-interse…

Chung–Goldwasser conjecture. Every measurable 3-sum-free subset $A$ of $[0,1]$ satisfies

Holroyd and Johnson's conjecture. If $\mathcal{A}\subseteq[n]^{(r)}_k$ is intersecting, then

Let $k\geq2$, $s\geq1$, $1\leq r\leq k-1$, and let $p\geq1$ be real. Suppose that … If $\mathcal F\subseteq\binom{[n]}k$ satisfies $\nu(\mathcal F)\leq s$, let $\mathcal B_S$ denot…

Let $1\leq t\leq k$, $1\leq r\leq k-1$, and let $p\geq2$ be real. Suppose that … If $\mathcal F\subseteq\binom{[n]}k$ is $t$-intersecting, then, for every $T\in\binom{[n]}t$, defin…

O'Neill's supersaturation conjecture. If $\mathcal{A}\subset 2^{[n]}$ consists of even-sized subsets and

Frankl's conjecture. For every IU-family $\mathcal F\subseteq 2^{[n]}$,

Let $n$, $k$, $t$, and $d$ satisfy … Let $\mathcal{F}\subseteq\binom{[n]}{k}$ be $t$-intersecting. Ellis–Keller–Lifshitz stability conjecture. If … then there exists a $t$-subset…

Balogh–Linz–Patkós' conjecture. There exists a positive integer $n_0=n_0(k,t)$ such that if $n+t$ is odd, $n>n_0$, and $\mathcal{F}\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sper…

Frankl–Wang's $3/7$ conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting, $n>100k$, and $|\mathcal{F}|>\binom{n-3}{k-3}$, then

Frankl–Kiselev–Kupavskii conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting and $n>10k$, then

Friedgut–Kahn–Kalai–Keller antipodal correlation conjecture. For every increasing $f,g:\{0,1\}^n\to\{0,1\}$, if $g$ is antipodal, then

Let $n\geq 2k>2\ell\geq 4$. Fix sets $F_0\in\binom{[2,n]}{k}$ and $G_0\in\binom{[2,n]}{\ell}$ with $F_0\cap G_0\neq\emptyset$, and define … … Two families are cross-intersecting if…

Frankl–Kleitman conjecture.

Let $V$ be a finite ground set, and let $\mathcal C$ be an intersection-closed family on $V$ containing $\varnothing$. Suppose that $\mathcal C$ admits a rooted-set representation…

Let $[n]=\{1,2,\ldots,n\}$, let $\mathscr{P}(n)$ be the power set of $[n]$, and let a family be an $(a,b)$-town modulo $k$ when all its sets have cardinality congruent to $a$ modul…

Let $G$ be a graph, let $calI^{(r)}(G)$ denote the family of independent $r$-subsets of $V(G)$, and let $calI^{(r)}_v(G)$ denote the star centered at $v\in V(G)$. Write $cmu(G)$ fo…

For fixed $k,\ell,s$, let $f(n,k,\ell,s)$ denote the maximum size of a family of $k$-element sets with no $\Delta(s+1)$-system whose kernel has size $\ell$. Let $\phi(\ell+1,s)$ be…

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote the power set of $[n]$, and let an IU-family be a family $\mathcal{G}\subseteq 2^{[n]}$ such that … and … Write $\beta(\mathcal{G})$…

Frankl's chain-free conjecture.

Chao–Xu–Yip–Zhang's conjecture.

For each triple $(n,k,\ell)$, let the right-hand side of be the lower-bound expression from Theorem, optimized over the parameter $j$. Asymptotic optimizer conjecture. For every…