56 problems

Lavrov–Loh conjecture. With probability $1-o(1)$ as $n\to\infty$, the random edge-ordering of $K_n$ contains a monotone Hamiltonian path.

Gishboliner–Jin–Sudakov conjecture. The exponent in this bound can be chosen to be linear in $k$.

A vertex-ordered forest is a forest whose vertices carry a linear order; its extremal function is the maximum number of vertices or edges avoiding a fixed ordered forest, and the i…

Algebraicity conjecture. Every $c\in\mathcal{S}$ is either an integer or an irrational algebraic number; equivalently,

No-accumulation-from-above conjecture. For every $c\in\mathbb{R}$, there exists $\varepsilon=\varepsilon(c)$ such that

Smallest-speed-above-2 conjecture. The smallest $c\in\mathcal{S}$ with $c>2$ is the largest real root of

Exponential growth-rate conjecture. The limit

Exponential–factorial jump conjecture. Either $|\mathcal{P}_n|<c^n$ for some constant $c$, or

Bradač–Liu–Wu–Xu conjecture. For every integer $k\geq 1$,

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 $K_a$ be a complete graph of order $a$ and $P_b^\mathrm{alt}$ an alternating path of order $b$. Complete-graph–alternating-path conjecture. For any $a \ge 2$ and $b \ge 3$, we…

Let $C_a^\mathrm{mon}$ be a monotone cycle of order $a$ and $P_b^\mathrm{alt}$ an alternating path of order $b$. Cyclic monotone-cycle–alternating-path conjecture. For any…

Let $C_a^\mathrm{mon}$ be a monotone cycle of order $a$ and $P_b^\mathrm{alt}$ an alternating path of order $b$. Monotone-cycle–alternating-path conjecture. For any $a \ge 3$ and…

Let $P_a^\mathrm{alt}$ be an alternating path of order $a$ and $P_b^\mathrm{ralt}$ a reverse alternating path of order $b$. Alternating–reverse-alternating conjecture. For any…

Let $P_a^\mathrm{alt}$ be an alternating path of order $a$ and $P_b^\mathrm{mon}$ a monotone path of order $b$. Mixed path conjecture. For any $a \ge 3$ and $b \ge 4$, we have … Th…

Let $P_a^\mathrm{mon}$ and $P_b^\mathrm{mon}$ denote monotone paths of orders $a$ and $b$, respectively, and let $R_\mathrm{cyc}$ be the cyclic Ramsey number. Cyclic monotone-path…

Non-nested matching Turán conjecture. The Turán number of non-nested $k$-matchings satisfies

The optimality conjecture. The lower bound is optimal for strongly non-nested matching, and hence also for non-nested matching:

Let $F$ be an ordered graph, meaning a graph equipped with a linear order on its vertices. An ordered forest is an ordered graph whose underlying graph is a forest, and the core of…

Gir27ao–Janzer–Janzer conjecture. The ordered Ramsey number of $P^<_{k,n}$ in $q$ colors admits the displayed bound with an exponent linear in $q$.

Let $f(k)$ be the Ramsey parameter defined in the source: the minimum parameter governing the relevant dependency-digraph Ramsey problem, and let $g(k)$ be the corresponding ordere…

Let $G$ be an edge-ordered graph, and let $sat_e(n,G)$ denote its edge-ordered saturation function. Almost-linear saturation conjecture. For every edge-ordered graph $G$, … This is…

Let $G$ be an edge-ordered graph, and let $sat_e(n,G)$ denote its edge-ordered saturation function. Near-linear saturation conjecture. For every edge-ordered graph $G$, … The paper…

Disjoint-neighborhood saturation conjecture. Under these hypotheses,

Let $G$) be an edge-ordered graph and let $e_0$ be an isolated minimal edge of $G$. Write $F=e_0+G$. The isolated-edge saturation conjecture. … The authors present this as a candid…