3 problems

Let $k$ be a positive integer, let $c_k=c(k)$ be an integer, and let $G$ be a graph on $n$ vertices such that $k\mid E(G)$. The composite-modulus linear-bound conjecture. There is…

Let $p$ be prime, let $c_p=c(p)$ be an integer, and let $G$ be a graph on $n$ vertices such that $p\mid E(G)$. The graph linear-bound conjecture. There is an integer $c_p=c(p)$ suc…

Let $p$ be prime, let $n_p=n(p)$ be a positive integer, and let $F$ be a forest on $n\geq n_p$ vertices such that $p\mid e(F)$. Here $R(F,\mathbb{Z}_p)$ denotes the least order of…