1. Show that |Hd con | ≤ 3d +1.

2. Conclude that VCdim(H) ≤ d log3.

3. Show that Hd con shatters the set of unit vectors {ei : i ≤ d}.

4. (**) Show that VCdim(Hd con) ≤ d.

Hint: Assume by contradiction that there exists a set C = {c1, . . ., cd+1} that is shattered by Hd con. Let h1, . . .,hd+1 be hypotheses in Hd con that satisfy ∀i , j ∈ [d +1], hi (c j ) = _ 0 i = j 1 otherwise For each i ∈ [d +1], hi (or more accurately, the conjunction that corresponds to hi ) contains some literal _i which is false on ci and true on c j for each j _= i. Use the Pigeonhole principle to show that there must be a pair i <>j ≤ d + 1 such that _i and _ j use the same xk and use that fact to derive a contradiction to the requirements from the conjunctions hi ,h j .