For the $k$-sparse set disjointness problem, where the parties each hold a $k$-element subset of an $n$-element universe, we show a tight ${\Theta(k \log k)}$ bound on the randomized one-way communication
complexity and also a slightly simpler proof of an $O(k)$ upper bound on the general randomized communication complexity of this problem, due originally to Hastad and Wigderson.

For the lopsided set disjointness problem, we obtain a simpler proof of Mihai's result, based on the information cost method. Our result shows that when Alice has a elements and Bob has b elements (${a < b}$) from an $n$-element universe, in any randomized protocol, either Alice must communicate ${\Omega(a)}$ bits of Bob communicate ${\Omega(b)}$ bits.