The sandwich is a coupling with explicit densities
Behague, Il'kovič and Montgomery, in a 2025 arXiv preprint last revised 8 December 2025, prove Kim and Vu's 2004 sandwich conjecture as Theorem 1.1: for each ε>0 there is a C>0 so that whenever the degree d is at least C log n, the random d-regular graph on n vertices can be coupled between two binomial graphs whose edge probabilities sit between (1−ε)d/n and (1+ε)d/n, with the regular graph contained in the denser one and containing the sparser one except on a vanishing-probability set.[1]
Quanta Magazine reported that proof on 18 September 2026. The study is still a preprint, not a journal article. The authors write that Gao, Isaev and McKay had already proved the sandwich for d much larger than log to the fourth of n, and that the coupling those authors introduced had been analysed only in a still denser window, d much larger than n over square-root log n.[1]
What the coupling actually moves
Take the regular graph apart as a middle slice. If a property survives adding or deleting a vanishing fraction of edges, it can be proved on the binomial graph, where the tools are older, and then read onto Gd(n) by containment. A property that is not stable under those edits, or that lives only on a vanishing set, does not ride the sandwich.[1]
That is a rebuilt machine, not a closed catalogue of networks. The authors analyse Gao–Isaev–McKay's existing coupling instead of inventing a third random-graph law. The remaining gap is which named properties anyone actually carries through the new log n window in a later paper.[1]
The next named transfer
The next check is a peer-reviewed journal version of this preprint, or a follow-up that cites Theorem 1.1 and names one binomial-graph property now proved for random regular graphs when d is only about log n. Until that arrives, the honest unit is the coupling theorem on arXiv, as Quanta described it.[1]