Re: Yes indeedy
I'm a mathematician, but this is based on a quick reading of this article alone so may be complete rubbish:
Imagine an endless list of random pluses and minuses.
Take any section of that list, or say every other symbol, or whatever kind of pattern you like from it. This gives you another list of pluses or minuses that you've plucked from the original list.
Work out whether you have more pluses than minuses in that or the other way around (or maybe an even number of both?). The difference is called the "discrepancy". A discrepancy of zero means there's the same number of pluses and minuses.
Using your (carefully-chosen) shorter list, and the discrepancy, you could then tell whether, for example, most of the pluses are in the beginning of your original list, or whether your list alternates between pluses and minuses, or whether it has a long run of pluses followed by a short run of minuses or whatever pattern you're looking for, just by looking at your short list extracted by a certain clever pattern. You can tell things about the infinite list just by carefully choosing the rule you use to extract the shorter list.
To translate the sentence: "For any sequence, Paul Erdős believed, you could find a finite sub-sequence that summed to a number bigger than any than you could choose – but he couldn't prove it."
What I think he's saying is, you can always find a smaller list inside that infinite list that - if you choose it carefully - has a discrepancy (i.e. more pluses or minuses) bigger than the original infinite list. So you could always "fudge" the numbers by misrepresenting the larger list with a carefully-chosen pattern.
But, to be honest, it's not entirely clear and probably a LOT more complicated than even the article makes out.
And I'd be hard-pushed to come up with something practical out of it (though I'm sure there would be - this is the sort of maths that sits behind things like coding theory and, thus, sending messages, compression, error-correction, RAID, etc.)