Quote:
Originally Posted by MindExodus
thus:
,
and can shatter all 3 point

MindExodus  very nice; I'm thoroughly convinced.
Quote:
Originally Posted by nkatz
Here is another example for the upper bound:
* H1 takes any 1 point and sets it to + or , with the other pointsif anygetting the opposite sign
* H2 sets all to + or all to 
H1 can only shatter 1 point, and H2 can only shatter 1 point
Their union however can shatter 3 points

nkatz, your example has me slightly puzzled. The only way H1 can fail to shatter 2 points is if there are exactly two points:
H1 contains + and +, but not  or ++
Whereas if you give it three points, it contains
+,
+,
+,
++,
++, and
++.
Just looking at the first two points, now H1 contains all 4 combinations. How can this be if its VC dimension is only 1? I think it must be the case that
, but I'd be unsurprised if I were missing something.