Q1. What is the consistency strength of the failure of square on singular cardinals?


Q2. What are known as partial results in this direction?



Consistency Strength of the Failure of Square on Singular Cardinals
One place to start reading is this:



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> James Cummings and Sy-David Friedman, "**[$\square$ On the Singular Cardinals](http://www.jstor.org/stable/27590333)**". *The Journal of Symbolic Logic* Vol. **73**, No. 4 (Dec., 2008), pp. 1307-1314.
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