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Claim. For a fixed translation-invariant linear relation, the theorem of J. Komlós, M. Sulyok and E. Szemerédi, Linear problems in combinatorial number theory, Acta Math. Acad. Sci. Hungar. 26 (1975), no. 1--2, 113--121, compares the largest relation-free subset guaranteed in every $n$-element set of integers, $g(n)$, with the largest relation-free subset of $\{1,\ldots,n\}$, $f(n)$: for all sufficiently large $n$, $g(n)\ge f(n)/(8\alpha^6)$, where $\alpha$ is the largest row $\ell^1$-norm of the relation's coefficients. Applied to the Sidon relation $a+b=c+d$, with $\alpha=4$ and $f(n)=(1+o(1))n^{1/2}$, it shows that every set of $N$ integers contains a Sidon subset of size at least $cN^{1/2}$. Points of the line have pairwise distinct distances exactly when they form a Sidon set, so in the notation of Problem 1088 $f_1(n)\le Cn^2$ for an absolute constant $C$. The paper states its theorem for integers; the transfer to finite sets of reals is recorded on [[problems/additive_bases/E0530/claims/1975_01_01_komlos_sulyok_szemeredi|the same theorem's claim page for Problem 530]], the problem the site names for $d=1$.

Covers. The upper bound $f_1(n)\ll n^2$. With the Erdős–Turán bound that a Sidon subset of $\{1,\ldots,N\}$ has at most $(1+o(1))N^{1/2}$ elements, which gives $f_1(n)\ge(1+o(1))n^2$, it fixes the order $n^2$ of the instance $d=1$. The constant is not determined.

Depends on. [[problems/additive_bases/E0530/claims/1975_01_01_komlos_sulyok_szemeredi|The claim page for Problem 530]], for the transfer of the integer theorem to finite sets of reals.

Acceptance. Refereed: the paper is a journal publication in Acta Mathematica Academiae Scientiarum Hungaricae, volume 26, issue 1--2 (1975). The site's remarks state $f_1(n)\asymp n^2$ through Problem 530, but the site labels the problem OPEN, so the remark is not reviewed evidence. The page is dated to the first day of the publication year, the citation giving no month or day.