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Progress on classifying small index subfactors has revealed an almost emptylandscape. In this paper we give some evidence that this desert continues up toindex 3+\sqrt{5}. There are two known quantum-group subfactors with index inthis interval, and we show that these subfactors are the only way to realizethe corresponding principal graphs. One of these subfactors is1-supertransitive, and we demonstrate that it is the only 1-supertransitivesubfactor with index between 5 and 3+\sqrt{5}. Computer evidence shows that anyother subfactor in this interval would need to have rank at least 38. We proveour uniqueness results by showing that there is a unique flat connection oneach graph. The result on 1-supertransitive subfactors is proved by an argumentusing intermediate subfactors, running the `odometer' from the FusionAtlas`Mathematica package and paying careful attention to dimensions.

The little desert? Some subfactors with index in the interval (5,3+\sqrt{5})