2,395 problems

Does every finite bridgeless graph have a collection of cycles in which every edge appears exactly twice?

Is the chromatic symmetric function $X_G(\mathbf x)$ Schur positive for every claw-free graph $G$?

At the conjectured density, must every $k$-uniform hypergraph contain a short nontrivial even cover, with constants free of superfluous polylogarithmic factors?

If a finite graph $G$ has girth at least five, must its minimum dual degree satisfy $\delta^*(G)\leq-\partial_n(G)$, where $\partial_n(G)$ is the smallest eigenvalue of its distanc…

For every connected graph, is the variance of its positive adjacency eigenvalues at most its order divided by its average distance?

Determine the Shannon capacities of odd cycles beyond $C_5$, or improve the best explicit independent-set bounds in their strong graph powers.

How sharply can elementary symmetric polynomials be bounded under vector or matrix centering constraints, and what quantitative consequences follow for permutation mixtures and fin…

For a finite family of distinct moduli $m<n_1<\cdots<n_k$ whose residue classes can be chosen pairwise disjoint, determine the largest possible reciprocal sum $\sum_i1/n_i$ as…

If every $m$-vertex subgraph is the union of an $r$-colourable graph and a graph with at most $f_r(m)$ edges, how large can $f_r$ be while forcing the whole graph to be $(r+1)$-col…

Must every $K_4$-free 4-chromatic graph contain an odd cycle with at least two diagonals? More generally, can local 3-colourability force odd cycles with arbitrarily many diagonals…

For every fixed $k\geq3$, does $R(k,l+1)/R(k,l)\to1$ as $l\to\infty$?

For every fixed $k\geq3$, is the off-diagonal Ramsey number $R(k,n)$ bounded below by $n^{k-1}/(\log n)^{c(k)}$?

For $k,r\geq2$, does there exist a set of integers with no nontrivial $(k+1)$-term arithmetic progression but whose every $r$-colouring contains a monochromatic $k$-term progressio…

Is there a constant $C$ such that every sufficiently large $A\subseteq[1,N]$ of size at least $5N/8+C$ contains distinct $a,b,c$ for which $a+b$, $a+c$, and $b+c$ also lie in $A$?

Compare maximal finite sets with at most $r$ representations of each sum to maximal sets with at most $r$ representations of each difference. Are their asymptotic constants unequal…

Does there exist an infinite-chromatic graph in which every $m$-vertex subgraph has an independent set of size at least $m/2-f(m)$ for some $f(m)\to\infty$?

If $A+A$ has positive upper density, can $A$ be split into $A_1\sqcup A_2$ so that both $A_1+A_1$ and $A_2+A_2$ have positive upper density? Is there a basis $A$ of order $2$ such…

Let $n_1<n_2<\cdots$ be such that, for any choice of classes $a_i\pmod{n_i}$, the uncovered integers have density zero. For every $\epsilon>0$, must some $k$ make the uncovered den…

For any $M\geq 1$, if $A\subset \mathbb{N}$ is a sufficiently large finite Sidon set, must there be at least $M$ sums $a\in A+A$ for which neither $a-1$ nor $a+1$ lies in $A+A$?

For a connected triangle-free graph $G$ on $n$ vertices, is there a constant $c>0$ such that fewer than $(1-c)n$ added edges always suffice to make the diameter 4 while keeping the…

How large can the clique-transversal number $\tau(G)$ be for an $n$-vertex graph? In particular, is $\tau(G)\leq n-\omega(n)\sqrt n$, or even $n-c\sqrt{n\log n}$?

For distinct real numbers $x_1,\ldots,x_n$, determine the maximum possible sum along a monotone subsequence.

For bases $3\leq d_1<\cdots<d_r$ satisfying the stated reciprocal-sum condition, can every sufficiently large integer be represented as a sum of distinct powers of the $d_i$? Under…

Let $H(n)$ be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than $n$. If $k_1=1$ and…

If $X_G$ is Schur positive, must $X_G(x_1,\ldots,x_k)$ have saturated Newton polytope for every finite $k$?