1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## 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 .

1. Show that |*Hd con *| ≤ 3*d *+1.

2. Conclude that VCdim(*H*) ≤ *d *log3.

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

4. (**) Show that VCdim(*Hd con*) ≤ *d*.

*Hint*: Assume by contradiction that there exists a set *C *= {*c*1*, . . ., **cd*+1} that is shattered by *Hd* *con*. Let *h*1*, . . .,**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 *.