16 problems

Let $C_n$ be the cycle graph on vertices $v_1,\ldots,v_n$, where $n\geq 4$ and each $v_i$ is adjacent to $v_{i+1}$ with indices taken modulo $n$. Let $\operatorname{TER}(C_n)$ deno…

Cyclic interval conjecture. The displayed inequality holds for every odd $m$ and every nonempty proper subset $x\subseteq\mathbb{Z}_m$; consequently,

Let $C_n^r$ denote the $r$th power of the cycle on $n$ vertices, and let $\Delta_d^t(C_n^r)$ denote its total $d$-cut complex. For $d\geq3$ and $n\geq(2d-1)r+1$, Shen et al.'s conj…

Let…

Let $C_6$ be the $6$-cycle, let $C_6(t)$ denote the associated $3$-uniform hypergraph obtained by adjoining a new vertex to every edge of $C_6$, and let $\operatorname{ex}(n,H)$ de…

Let $C_k$ denote the cycle graph on $k$ vertices, and let $\mathrm{sat}(n,C_k)$ be its saturation number. Füredi–Kim conjecture. There exists a constant $k_0$ such that … holds for…

Online cycle Ramsey conjecture.

Let $C_{2k}$ be an even cycle and let $C^d_{2k}$ denote the $d$-fold Cartesian product of $C_{2k}$ with itself. A distance-three matching is a matching whose distinct edges are at…

Let $C_8$ be the cycle on eight vertices, and let $\operatorname{ex}(n,H)$ denote the maximum number of edges in an $H$-free graph on $n$ vertices. $C_8$ extremal-number conjecture…

Spectral cycle-length conjecture. The largest such constant $C$ is characterized by the property that every such graph contains a cycle of every length $t\le (C-\varepsilon)n$. The…

Induced 4-cycle construction conjecture. For every real number $x\in[1/2,1]$,

The anti-directed cycle Turán conjecture. For every $k\in \mathbb{N}^*$, there exists a constant $n_0$ such that for $n\ge n_0$,

Let $C_4$ denote the cycle of length four, and let $\operatorname{ex}^{-1}(k,H)$ denote the inverse Turán number of $H$, namely the maximum number of edges in a graph whose every…

Let $C_t$ be the cycle on $t$ vertices, and let $f_{C_t}(s,s)$ denote the least number of independent sets of size $s$ in $C_t$ that guarantees a rainbow independent set of size…

Cycle-union conjecture. The union

Nonexistence conjecture. For $t\ge 6$, there are no nontrivial uniquely $C_t$-saturated graphs.