GPT 6 Astra Ultra attempt on Math-5451
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FORMAL RESTATEMENT
Erdős #134; reconstruction, 2026-09-23. Given $\varepsilon,\delta>0$, is it true that, for all sufficiently large $n$, every triangle-free $n$-vertex graph with maximum degree below $n^{1/2-\varepsilon}$ can be extended on the same vertex set to a triangle-free graph of diameter $2$ by adding at most $\delta n^2$ edges?QUICK LITERATURE AND CONTEXT CHECK
The current page records a positive answer. Alon's primary note proves it by a bounded-degree random extension followed by maximal completion. We reconstruct that proof with $\eta=\min(\varepsilon,1/12)$, sufficient for the question, and supply the relevant probability comparison. No new bound or novelty is claimed.WORK
Set $c=n^{-\eta}$, $D=\lceil2c\sqrt n\rceil$, and $m=\lfloor c^2 n^{3/2}\rfloor$. Starting from the given graph, make $m$ random additions. At each step choose uniformly from the pairs of nonadjacent vertices that both have degree below $D$ and have no common neighbor. Adding such an edge preserves triangle-freeness and the degree bound $D$.At any stage at most $2cn$ vertices have reached degree $D$: each such vertex has acquired at least $c\sqrt n$ new incidences, while there are at most $2m$ new incidences in total. The number of unordered pairs having a common neighbor is at most \[ \sum_v\binom{d(v)}2\le nD^2/2=(2+o(1))c^2n^2. \tag{1} \] There are at least $\binom{n-2cn}{2}$ pairs of unsaturated vertices, and at most $nD/2=O(cn^{3/2})$ existing edges. These estimates ensure an eligible pair always exists for large $n$, so the process can perform all $m$ steps.
Fix a set $U$ of $u=\lceil5cn\rceil$ vertices. Conditional on $U$ still being independent, at least $3cn$ of its vertices are unsaturated. By $1$, its eligible internal pairs number at least \[ \binom{3cn}{2}-(2+o(1))c^2n^2\ge2c^2n^2 \] for large $n$. There are at most $n^2/2$ eligible pairs altogether. Thus, at each step before $U$ ceases to be independent, the chance of selecting an edge inside it is at least $4c^2$, and in particular at least $3c^2$. Iterated conditional probabilities give \[ \Pr(U\hbox{ remains independent})\le(1-3c^2)^m \le\exp(-3c^2m). \] The union bound over all $U$ gives probability at most \[ \exp\left(u\log(en/u)-3c^2m\right). \tag{2} \] The positive term is $O(cn\log n)$, whereas the magnitude of the negative term is $(3+o(1))c^4n^{3/2}$. Their ratio tends to infinity, since \[ \frac{c^3\sqrt n}{\log n} =\frac{n^{1/2-3\eta}}{\log n}\longrightarrow\infty. \] Therefore $2$ is less than one for large $n$. There is an extension $H$ with independence number $\alpha(H)<u$.
Now add edges to $H$ as long as this preserves triangle-freeness, obtaining a maximal triangle-free graph $J$. Every neighborhood in $J$ is independent in $J$, and hence independent in $H$. Thus \[ \Delta(J)<u,\qquad e(J)\le n(u-1)/2=O(cn^2)=o(n^2). \] Every nonadjacent pair of vertices in $J$ has a common neighbor; otherwise their edge could still be added. Consequently $J$ is connected and has diameter at most $2$. For $n\ge3$ a triangle-free graph cannot be complete, so its diameter is exactly $2$. The number of edges added to the original graph is at most $e(J)=o(n^2)$, and is therefore at most $\delta n^2$ for all sufficiently large $n$.$\square$