Every Seven Positive Reals Contain a Dissociated Four-Element Subset: Small Cases of an Erdős Problem on Dissociated Sets

Let \(f(n)\) be the largest \(k\) such that every set of \(n\) real numbers contains a dissociated subset of size \(k\), that is, a subset whose \(2^k\) subset sums are pairwise distinct (the empty sum included). Erdős noted that \(f(n)\ge\lfloor \log_3 n\rfloor\) and asked whether \(f(n)\ge\lfloor \log_2 n\rfloor\) for every \(n\) (Erdős Problem #963). The general question remains open. Let \(g^{+}(K)\) be the largest size of a set of positive reals without a dissociated subset of size \(K\). We show that the corresponding largest size for sets of reals is \(1+2g^{+}(K)\), so that the question is equivalent to \(g^{+}(K)\le2^{K-1}-1\) for all \(K\), and also to the inequality \(|P|\le2^{\dim P}-1\) for finite sets \(P\) of nonzero integer vectors containing no pair \(x,-x\), where \(\dim P\) is the largest size of a dissociated subset of \(P\). Elementary arguments give \(g^{+}(1),g^{+}(2),g^{+}(3)=0,1,3\). Our main result is \(g^{+}(4)=6\): every seven positive reals contain a dissociated four-element subset, and \(\{1,\dots,6\}\) contains none. The proof is computer-assisted and is given twice: by a case analysis over sign patterns in which every terminal case is closed by an exact Farkas certificate, checked by independent exact checkers, and by an enumeration of linear forms over a dissociated triple in exact integer arithmetic. Consequently \(f(n)\) is known for all \(n\le27\): it is \(0,1,1,2,2,2,2\) for \(n\le7\), then \(3\) for \(8\le n\le13\) and \(4\) for \(14\le n\le27\); and \(f(n)\ge\lfloor \log_2 n\rfloor\) holds for all \(n\le31\). The first open case, \(n=32\), is equivalent to \(g^{+}(5)\le15\). The upper bound for \(14\le n\le27\) uses the set \(\{1,\dots,10,12,13,15\}\) without dissociated five-element subsets, posted by BAKKAOUI on the problem's forum (2026); we also show \(g^{+}(6)\ge24\) by an explicit set. An unrefereed argument for \(f(n)\ge(1-o(1))\log_2 n\) was posted on the same forum by KoishiChan (2025). To our knowledge the value \(g^{+}(4)=6\) and its consequences for \(n\le31\) are new. This is an unrefereed note.