There are several synonyms for the phrase "gives the high sign," which means to signal someone to do something. One such synonym is "gives the go-ahead," which means to authorize or approve of something. Another synonym is "gives the nod," which means to indicate agreement or approval. "Signals approval" is another phrase that can be used instead of "gives the high sign." Additionally, "gives the signal" or simply "signals" can also be used to convey the same meaning. No matter which synonym is used, the intent behind the phrase remains the same: to indicate that it is okay to proceed with a certain action.