Brockman–Kay conjecture on intersecting families of multisets

Let $[n]=\{1,\ldots,n\}$, let $M(n,k)$ be the $k\times n$ rectangle representation of $k$-multisets, and let $\mathcal{M}(n,k,t)$ denote the families of $t$-intersecting $k$-multisets of $[n]$. Brockman–Kay conjecture. There is $n_0(k,t)$ such that if $n\geq n_0(k,t)$ and $\mathcal{F}\in\mathcal{M}(n,k,t)$, then

$$|\mathcal{F}|\leq\binom{n+k-t-1}{k-t}.$$

Furthermore, equality is achieved if and only if every member of $\mathcal{F}$ contains a fixed $t$-multiset of $M(n,k)$. This conjecture concerns the extremal size of $t$-intersecting multiset families; the paper records a later result establishing the bound under the explicit threshold $n\geq t(k-t)+2$, while the equality characterization is given in the corresponding threshold conjecture.