  Set-disjointness problems are one of the most fundamental problems in
communication complexity and have been extensively studied in past decades.
Given its importance, many lower bound techniques were introduced to prove
communication lower bounds of set-disjointness. Combining ideas from
information complexity and query-to-communication lifting theorems, we
introduce a density increment argument to prove communication lower bounds for
set-disjointness:
  We give a simple proof showing that a large rectangle cannot be
$0$-monochromatic for multi-party unique-disjointness.
  We interpret the direct-sum argument as a density increment process and give
an alternative proof of randomized communication lower bounds for multi-party
unique-disjointness.
  Avoiding full simulations in lifting theorems, we simplify and improve
communication lower bounds for sparse unique-disjointness.
  Potential applications to be unified and improved by our density increment
argument are also discussed.
