Section 4 — an adversary forces $kr$ sets on the block family while $\mathrm{OPT} = 1$

Let $k, r$ be positive integers and let $\mathcal F$ be the block family of Section 4 on $kr^2$ disjoint blocks of $2^k$ elements. For every valid deterministic online algorithm $A$ for $\mathcal F$ there is a nonempty arrival sequence $\sigma$ of at most $kr$ elements such that

1. a single set of $\mathcal F$ contains every element of $\sigma$, so $\mathrm{OPT}(\sigma) = 1$, and
2. the algorithm has chosen at least $kr$ sets:

This is the paper's claim that "given any deterministic algorithm, an adversary can choose $kr$ elements in $X$, forcing the algorithm to pick $kr$ sets from $\mathcal F$, while keeping the value of the optimum solution to be 1". It is the core of Proposition 4.2, which adds padding elements and sets to reach arbitrary $n$ and $m$.

Formalization Note The paper's argument tacitly lets the algorithm add one set per arrival; here algorithms may add any number of sets per arrival, and the sequence then may be shorter than $kr$, hence "at most $kr$ elements". The ground set is the blocks only; elements outside the blocks lie in no set and play no role.