  In this article we investigate a problem in graph theory, which has an
equivalent reformulation in extremal set theory similar to the problems
researched in "A general 2-part Erd\H{o}s-Ko-Rado theorem" by Gyula O.H.
Katona, who proposed our problem as well. In the graph theoretic form we
examine the clique number of the Xor product of two isomorphic $KG(N,k)$ Kneser
graphs. Denote this number with $f(k,N)$. We give lower and upper bounds on
$f(k,N)$, and we solve the problem up to a constant deviation depending only on
$k$, and find the exact value for $f(2,N)$ if $N$ is large enough. We also
compute that $f(k,k^2)$ is asymptotically equivalent to $k^2$.
