Discoveries / combinatorics

Ramsey-Style Hypergraph Partition Bound H(n)

Let $H(n)$ be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than $n$. With $k_1 = 1$ and $k_n = \lfloor n/2 \rfloor + k_{\lfloor n/2 \rfloor} + k_{\lceil n/2 \rceil}$, prove $H(n) \ge c\,k_n$ for some constant $c > 1$, already for $n = 15$, with a constructive algorithm.