5 problems

Krivelevich–Sudakov conjecture. There exists $C>0$ such that if

Let $G$ be an $(n,d,\lambda)$-graph, meaning an $n$-vertex $d$-regular graph whose non-trivial eigenvalues have absolute value at most $\lambda$. A Hamilton cycle is a cycle contai…

Let $\Delta\geq2$. An $(n,D,\lambda)$-graph is a graph with the corresponding order, degree and spectral parameters, and a $[t]$-colored tree is a tree whose edges receive colors i…

For $k\geq 2$, let an $(\varepsilon,p,k-1,k)$-pseudorandom graph mean a graph satisfying the stated pseudorandomness condition with parameters $(\varepsilon,p,k-1,k)$. The $k$th po…

Global resilience conjecture. Then $G$ has global resilience $(1/4+o(1))nd$ with respect to being $C_k$-free.