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Claim. P. Turán, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436--452 (in Hungarian; Zbl 0026.26903), the paper that footnote 1 on p. 122 of Erdős's On a theorem of Rademacher-Turán (Illinois J. Math. 6 (1962); filed as erdos_1962_theorem_rademacher_turan) cites for Turán's theorem. As the zbMATH review of Zbl 0026.26903 states it, the paper proves that among graphs on $n$ vertices containing no complete graph on $k$ vertices, the graph $D(n,k)$ has the maximum number of edges, and is the only such graph; here $D(n,k)$ joins two vertices exactly when their labels in $1,\dots,n$ are incongruent modulo $k-1$, and it has $\frac12(k-2)(k-1)^{-1}(n^2-r^2)+\binom r2$ edges, where $r$ is the remainder of $n$ on division by $k-1$. The case $k=3$, which Mantel proved in 1907, is the statement that p. 122 of Erdős's paper records as "a special case of Turán's theorem": a graph on $n$ vertices with more than $\lfloor n^2/4\rfloor$ edges contains a triangle. The extremal graph $D(n,3)$ is the complete bipartite graph $K_{\lfloor n/2\rfloor,\lceil n/2\rceil}$, which has $\lfloor n^2/4\rfloor$ edges, no triangle and, for $n\ge2$, chromatic number $2$. So, in the conventions of Problem 1011, $f_2(n)=\lfloor n^2/4\rfloor+1$ for every $n\ge2$; Erdős's 1971 list gives the same value as $u_2=[\tfrac14n^2]+1$, "the well known theorem of Turán". The record gives the year only, so the page's month and day are placeholders.

Covers. The case $r=2$: $f_2(n)=\lfloor n^2/4\rfloor+1$ for every $n\ge2$, the condition being vacuous for $n=1$. Nothing about $r\ge3$.

Depends on. Nothing in this wiki; Turán's theorem is the whole argument.

Acceptance. Refereed journal publication in Matematikai és Fizikai Lapok, which is the refereed evidence. The site credits Turán's theorem on a problem it labels OPEN, which is not acceptance, so reviewed is not listed. The paper is in Hungarian and is not examined in this corpus; the statement is taken from the zbMATH review of Zbl 0026.26903, and Erdős's 1962 and 1971 papers, which quote the theorem, agree with it.