## Unique one-loop effective action for the six-dimensional Einstein-Hilbert action

In this Letter we give the gauge-independent divergent part of the one-loop correction to the effective action for the six-dimensional Einstein-Hilbert action, including a cosmological constant. We extend the methods of Barvinsky-Vilkovisky from four to higher dimensions, including the Vilkovisky-DeWitt correction, to obtain the unique counterterms for an arbitrary background field. All one-loop operators are given for {ital N} dimensions; however, due to the complexity of the Schwinger-DeWitt expansion, going beyond six dimensions to eight or ten will probably require restricting the background.