Kang–Nikiforov–Yuan conjecture on the spectral radius of chromatic hypergraphs

Let $G$ be an $r$-uniform hypergraph, and let $\binom{n}{r}$ denote the number of $r$-subsets of an $n$-vertex set. For $p\geq 1$, let $\lambda^{(p)}(G)$ be the maximum of the polyform of $G$ over the unit sphere in the $\ell_p$-norm. An $r$-graph is $k$-chromatic if its vertex set can be partitioned into $k$ classes so that no edge is contained in a single class. For nonnegative integers $n_1+\cdots+n_k=n$, let $Q(n_1,\ldots,n_k)$ be the complete $k$-chromatic $r$-graph with color classes of sizes $n_1,\ldots,n_k$, and let $Q_k^r(n)$ be the member whose color-class sizes differ by at most one. Kang–Nikiforov–Yuan's conjecture. Let $k\geq 2$, $r\geq 4$, and let $G$ be a $k$-chromatic $r$-graph of order $n>(r-1)k$. For every $p\geq 1$,

$$\lambda^{(p)}(G)<\lambda^{(p)}(Q_k^r(n)),$$

unless $G$ is isomorphic to $Q_k^r(n)$. Kang–Nikiforov–Yuan established the corresponding exact theorem for $3$-graphs; the conjecture extends their $p$-spectral result to general uniformity and asserts uniqueness of the balanced complete $k$-chromatic hypergraph.