We study pairs of families ${\cal A},{\cal B}\subseteq
2^{\{1,\ldots,r\}}$ such that $|A\cap B|\in L$ for any
$A\in{\cal A}$, $B\in{\cal B}$. We are interested in the maximal
product $|{\cal A}|\cdot|{\cal B}|$, given $r$ and $L$. We give
asymptotically optimal bounds for $L$ containing only elements
of $s<q$ residue classes modulo ... more >>>

Razborov~\cite{Razborov96} recently proved that polynomial
calculus proofs of the pigeonhole principle $PHP_n^m$ must have
degree at least $\ceiling{n/2}+1$ over any field. We present a
simplified proof of the same result. The main
idea of our proof is the same as in the original proof
of Razborov: we want to describe ... more >>>

We introduce the model of conservative one-way multiparty complexity
and prove lower and upper bounds on the complexity of pointer jumping.

The pointer jumping function takes as its input a directed layered
graph with a starting node and layers of nodes, and a single edge ... more >>>

We prove an unexpected upper bound on a communication game proposed
by Jeff Edmonds and Russell Impagliazzo as an approach for
proving lower bounds for time-space tradeoffs for branching programs.
Our result is based on a generalization of a construction of Erdos,
Frankl and Rodl of a large 3-hypergraph ... more >>>