Infinite words containing squares at every position
  Richomme asked the following question: what is the infimum of the real numbers $\alpha$ > 2 such that there exists an infinite word that avoids $\alpha$-powers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is $\alpha$ = 7/3. 
James D. Currie, Narad Rampersad