As revealed in later editions of yesterday's Evening Times, the fight happened in a car park near shops in Hamilton's Mill Road at about 8am yesterday.
Two men, aged 20 and 22, were taken to Hairmyres Hospital in East Kilbride where they are in a serious but stable condition. It is believed they were stabbed.
A third man, aged 30, was taken to Wishaw Hospital. He was released after treatment.
Detective Inspector David Tassie said: "We do not believe that this was a random assault.
"I would appeal to anyone who may have information to contact us on 01698 483 300 or Crimestoppers on 0800 555 111."