 The rivalry intensified in 1941 when Ted Williams of the Red Sox batted .406 , becoming the last player to bat over .400 in a season . Despite his accomplishment , Williams lost the AL MVP race to the Yankees ' Joe DiMaggio , who holds the record for a hitting streak , with 56 straight games in which he had a hit . DiMaggio 's unprecedented streak started on May 15 , 1941 in a game where the Yankees lost 13 – 1 to the Chicago White Sox . Williams later reminisced about his rivalry with DiMaggio saying " ( Joe ) DiMaggio was the greatest all @-@ around player I ever saw . His career cannot be summed up in numbers and awards . It might sound corny , but he had a profound and lasting impact on the country . " Both teams almost swapped the two players . In 1947 , Boston Red Sox owner Tom Yawkey and Yankees GM Larry MacPhail were rumored to have verbally agreed to trade DiMaggio for Williams , but MacPhail refused to include Yogi Berra and the deal fell through . Joe DiMaggio 's younger brother Dom would play for the Red Sox his entire career during the 1940s . 
