are some striking applications in which expanders yield good results but not good enough. For example, by the celebrated result of Sipser and Spielman [@SS96], expanders imply good error-correcting codes. However, the codes obtained by [@SS96] are not locally testable, namely, the very desired property of local testability (which in fact was the motivating goal in the construction of error-correcting codes from expanders, as discussed in Spielman’s thesis [@Spi95]) is not known to be achieved by codes based on expanders.

Another canonical example where we have seen the limit of expanders is the new proof of the PCP theorem obtained by Dinur [@Din07]. Dinur’s proof builds on properties of expanders to yield a simpler proof of the celebrated PCP theorem. However, the PCP obtained by expanders is not strong enough to yield good inapproximation results.

The thesis we try to pursue here is that high dimensional expanders could potentially be used in applications as above, where (one-dimensional) expanders were not strong enough. We expect high dimensional expanders to yield better results than the well studied (one-dimensional) expanders since they are stronger objects: They expand at [*all*]{} dimensions, i.e., with respect to vertices, edges, triangles, etc., and not just with respect 