In this work, using methods from high dimensional expansion, we show that the property of k -direct-sum is testable for odd values of k . Previous work of Kaufman and Lubotzky could inherently deal only with the case that k is even, using a reduction to linearity testing. Interestingly, our work is the first to combine the topological notion of high dimensional expansion (called co-systolic expansion) with the combinatorial/spectral notion of high dimensional expansion (called colorful expansion) to obtain the result.

The classical k -direct-sum problem applies to the complete complex; Namely it considers a function defined over all k -subsets of some n sized universe. Our result here applies to any collection of k -subsets of an n -universe, assuming this collection of subsets forms a high dimensional expander.