Size-4 Counterexamples to the Sidon-Extension Conjecture

A finite set $S\subset \mathbb{Z}$ is a Sidon set if its pairwise differences are distinct. Recall that a perfect difference set (PDS) of order $n$ is a set $B\subset {\mathbb{Z}}_v$ ($v=n^2-n+1$) of size $n$ such that every nonzero residue arises exactly once as a difference of two elements of $B$. Erdős’s $1000 conjecture – that every finite Sidon set extends to a finite PDS – was disproved by Alexeev and Mixon (arXiv:2510.19804, October 2025), via the size-5 counterexamples $\{1,2,4,8,13\}$ and Hall’s earlier $\{1,3,9,10,13\}$; they then asked: what is the smallest size $s$ of a non-extending Sidon set? The trivial bounds give $3\le s\le 5$. Our evidence points to $s=4$. We exhibit two integer Sidon sets, $$A=\{0,1,3,11\},B=\{0,1,4,11\},$$ together with the apparent infinite family of dilations $kA$, $kB$ and their reflections, all of which fail to extend for every prime power $q\le 317$ via the Singer affine-orbit check (rigorous under Hall’s 1947 uniqueness for Desarguesian cyclic planes through $q\le 40$ and under the prime-power conjecture beyond that), and unconditionally for every modulus $v\le 133$ via brute-force depth-first search. We also report the exact density $N_{\text{ne}}(N)=4\lfloor N\mathrm{/}11\rfloor$ of non-extending size-4 Sidon sets in $[0,N]$ for $N\le 50$ – the match is exact, which suggests the $kA,kB$ family is complete in this range. A complete proof, perhaps in the spirit of Alexeev–Mixon’s polarity argument or via a multiplier descent, remains open.