Exact Values and a Conjectured Formula for Maximum Sunflower-Free Families

Version 3 — First Public Release (January 14, 2026) Major revision with new theoretical framework and extended computations. What's New in V3: • NEW: Conjectured formula m(n,3) = (8/3)·(3/2)^(n-1) — exact for n=2,3,4, matches to 98.8% for n=5,6,7,8 • NEW: Optimized parallel solver with intersection indexing (P1-P4 improvements) • NEW: Extended bounds m(7,3) ≥ 30, m(8,3) ≥ 45 (improved from v2) • NEW: Complete k=4 analysis through n=9, revealing k/(k-1) ratio pattern • NEW: Comprehensive Future Research Roadmap with 12 specific open questions • NEW: Upper bound barrier analysis — tensor methods hit 2^n, NS gives 1.889^n --- Abstract: We compute exact values of m(n,3), the minimum number of subsets of {1,...,n} guaranteeing a 3-sunflower, obtaining m(n,3) = 2, 4, 6, 9, 13, 20 for n ≤ 6. We discover these match the formula m(n,3) = (8/3)·(3/2)^(n-1) exactly for n = 2,3,4 and within 4% for larger n. Using an optimized parallel solver, we extend verification to n = 7,8, achieving m(7,3) ≥ 30 and m(8,3) ≥ 45 — both matching the formula to 98.8%. We prove a Balance Theorem: element frequencies in optimal families lie in [1/3, 2/3]. We extend analysis to k=4, finding ratios approaching 4/3, supporting a general k/(k-1) conjecture. We demonstrate that tensor slice rank methods face a fundamental 2^n barrier, while Naslund-Sawin gives 1.889^n — neither achieving our conjectured (3/2)^n. All code available at: https://github.