An EKR Theorem for the Cartesian Product of Complete Graphs

The Erdős-Ko-Rado theorem states that for r ≤ n/2, the largest intersecting family of r-subsets of $[n]$ is given by fixing a common element in all subsets, which trivially ensures pairwise intersection. We investigate this property for families of independent sets in the Cartesian product of complete graphs, Kₙ × Kₘ. Using a novel extension of Katona's cycle method, we prove Kₙ × Kₘ is r-EKR when 1 ≤ r ≤ min(m,n)/2, demonstrating the Holroyd--Talbot conjecture holds for this class of well-covered graphs.