The assistant chief constable told Reuters in an interview that police were taking nothing for granted ahead of Sunday's quarter-final with Italy, and they would make special provision if England reached a semi-final against Germany in Warsaw.
The uniformed officer, speaking outside the giant fan zone in central Kiev, said police expected about 6,000 fans for Sunday's England match, the biggest number so far following their group games against France, Sweden and Ukraine.
"There have been no arrests of England fans I am pleased to say. We have worked very hard with both the embassy and consular services and are very happy.
"Of course, UEFA fined the English FA for the fans behaviour at the Sweden game, but that was inside the stadium not outside and there have been no incidents outside."
"We have also worked very hard with the Ukrainian authorities and UEFA itself to get even better policing and stewarding at the ground but, overall, there have been no real problems."
The English FA was fined 5,000 euros after England fans tried to invade the Kiev pitch during the 3-2 win over Sweden on June 15 in their last Group D game.
If England reach the semi-finals, plans are already in place to deal with what would be a high profile match against Germany, he explained.
"We have a senior office in Poland already working with the stadium and Polish police authorities planning for that match if it should take place.
"There will be a reinforced and enhanced capability in Poland because we would anticipate a significant increase in the number of England fans at that match."
England's travelling army, "a very different breed of fan from 15 or 20 years ago," according to Holt, are being marshalled by officers in uniform and plainclothes who, Holt says, have a very good relationship with the fans.
He added that there have been some isolated cases in Ukraine where England supporters had been the victims of crime which had been dealt with.
"One England fan, an off-duty police officer on holiday, was assaulted in Donetsk," he said, "but so far there have not been too many other incidents and generally we are very happy at what has been a largely trouble-free tournament so far."
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