Best communication complexity lower bound of disjointness
In a recent paper, Braverman, Garg, Pankratov, and Weinstein compute the value of $\delta$ to be exactly some constant around 0.4827, up to sublinear factors. This gives a tight bound on the communication complexity of disjointness.

The constant itself was found using a computer algebra system, and as far as I'm aware can't be expressed simply.