Can I answer your first question this way? (I would like to know if there is any mistake in the approach I take below.)

So, you represent a set by its characteristic vector which are bitstrings of length each containing an odd number of ‘s. I assume that you can get such ‘s which have odd number of elements and the intersection of and contains even number of elements.

Like you have done above, let us view this as a vector space over . Now, if it is possible for me to stuff more than number of sets, let us look at as a linear combination of minimal number of basis elements (which has fewer than ) elements. Then, if , then is 1 for all (by minimality).(Here is XOR) and since intersection is even, the value should be be true for XOR which is not the case. Contradiction => m is at most n. And m = n is possible. So, the result follows.

]]>Let S be a finite set and let T be a proper nonempty subset of S. Suppose R is selected uniformly at random from among all subsets of S of odd size. what is the probability that size of (T intersection R) is odd?

any ideas?

]]>Here is an exercise that is related to Example 2 which you may find interesting. Fix k and let n >>k. How many subsets of {1,..,n} can one find such that each has even size and every k-intersection also have even size ?

]]>The proof in your post seems more natural than the one in the book of van den Dries (which is essentially the same as that of Vapnik-Chernovenkis), by induction on k. ]]>