The explicit p-spectral Turán conjecture for k-chromatic uniform hypergraphs

Let $G$ be a $k$-chromatic $r$-graph of order $n$, where $k\geq 2$ and $n>(r-1)k$. For $p\geq 1$, let $\lambda^{(p)}(G)$ be the $p$-spectral radius of $G$, and let $Q_k^r(n)$ be the complete $k$-chromatic $r$-graph of order $n$. The explicit p-spectral Turán conjecture. For every $p\geq 1$,

$$r!\left(\binom{n}{r}-k\binom{n/k}{r}\right)n^{-r/p}$$

is a strict upper bound for $\lambda^{(p)}(G)$, unless $k$ divides $n$ and $G=Q_k^r(n)$. This is an explicit strengthening of the preceding extremal conjecture, giving a closed-form bound in the range where the authors state that their methods do not provide the generalization.