97 problems

Associated Hermite matching-generating-function conjecture. The integral

Let $S$ be an even set of points in the plane, and let $\mathcal M$ be a max-sum matching of $S$. Define $t(S)$ as the minimum, over points $y$ in the plane, of…

Let $S$ be an even set of points in the plane, and let $\{a_ib_i:i=1,\dots,n\}$ be a max-sum matching of $S$, where $|S|=2n$. A point $o$ is required to satisfy, for every matched…

Let $G$ be an $r$-regular graph with maximum degree $\Delta$ and edge-chromatic number $\chi'(G)=\Delta+1$. A maximum $\Delta$-colorable subgraph is a subgraph with as many edges a…

Let $G$ be a finite, simple, connected, $\Delta$-regular graph of order $n$, where $\Delta\geq 3$. The edge domination conjecture asserts that … Equality holds if and only if $G$ h…

Mkrtchyan–Petrosyan–Vardanyan conjecture. If

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…

Let $T$ be a tournament, namely an oriented graph in which every pair of distinct vertices is joined by exactly one directed arc. For a vertex $x$, let $N_T^+(x)$ and $N_T^{++}(x)$…

Let $D$ be an oriented graph. For a vertex $x$, let $N_D^+(x)$ and $N_D^{++}(x)$ denote its out-neighborhood and second out-neighborhood, respectively. A complete matching from $X$…

Let $M_s^r$ be a matching of size $s$ in an $r$-uniform hypergraph, and let $\operatorname{ex}_r(n,M_s^r)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergrap…

Type $(4,3)$ finite asymptotic separation index matching conjecture. If $\mathsf{asi}(G)<\infty$, then $G$ has a Borel matching covering $A$.

Finite asymptotic separation index matching conjecture. If the bipartition has combinatorial expansion and $\mathsf{asi}(G)<\infty$, then $G$ has a Borel matching covering $A$.

Erdős–Kleitman matching conjecture. Suppose that $s\ge 2$, $m\ge 1$, and $n=sm+s-\ell$ for some integer $0<\ell\leq\lceil s/2\rceil$. Then

Engbers–Erey's conjecture. The average size of maximal matchings, $avm(T)$, is uniquely minimized by

Let $G$ and $H$ be graphs, and let $\mathcal{R}(G,H)$ denote the family of Ramsey-minimal graphs for the pair $(G,H)$. A matching is a graph whose connected components are all isom…

Let $G$ and $H$ be graphs, and let $K_2$ denote the complete graph on two vertices. A pair $(G,H)$ is Ramsey-finite if it has only finitely many Ramsey-minimal graphs. Burr–Erdős–F…

Let $M_a^\mathrm{nest}$ be a nested matching with even parameter $a$, and let $P_b^\mathrm{mon}$ be a monotone path of order $b$. Nested-matching–monotone-path conjecture. For any…

Let $M_a^\mathrm{nest}$ and $M_b^\mathrm{nest}$ denote nested matchings with parameters $a$ and $b$. Nested-matching conjecture. Let $a, b \ge 4$ be divisible by four. Then … A con…

Constant-deviation conjecture. There exists an absolute constant $C$ such that for all positive integers $k$ and $t$, every colour-balanced $k$-edge-colouring of $K_{2kt}$ admits a…

Square-root discrepancy conjecture. There is a perfect matching $M$ of $K_{n,n}$ satisfying

Pardey and Rautenbach's conjecture. For all integers $k\geq 2$ and $t\geq 1$, every colour-balanced $k$-edge-coloured $K_{2kt}$ admits a perfect matching $M$ satisfying…

Let $n$ be even. In the rainbow perfect matching game $\mathcal{RP}_n$, played on $n/2$ copies of $K_n$, Maker wins by claiming a rainbow perfect matching; let $b_{\mathcal{RP}_n}$…

Let $q,s\ge 1$ and $r\ge 2$ be integers. An $s$-connector is an $r$-uniform hypergraph such that every collection of pairwise disjoint vertex sets $A_1,\dots,A_r$ with $|A_i|\ge s$…