# Number of $0/1$-monochromatic rectangles and communication complexity

What is the relation between number $0$-monochromatic rectangles in characteristic matrix and communication complexity?

What is the relation between number $1$-monochromatic rectangles in characteristic matrix and communication complexity?

Precisely how does this connect to logrank conjecture?

Is the number of $0$-monochromatic rectangles always almost same as $1$-monchromatic rectangles?