Halls theorem — the series
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One bottleneck and nothing else
A set of jobs can be filled by distinct people unless some group of jobs has too few candidates between them — and that single obstruction is the only one there is, which is what makes the theorem worth having.
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What the search has when it fails
A largest matching is easy to find and hard to certify: the claim that nothing larger exists is a claim about every arrangement not tried. The certificate turns out to be free — it is the wreckage of the search that failed.
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The piece that cannot pair off
Take the sides away and the obstruction to a matching changes character completely. It is no longer a shortage of partners; it is a parity, and the quantity that measures it counts pieces of odd size rather than vertices of any size.
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Enough partners in every finite group
Hall's theorem says a matching exists unless some group has too few partners between them. On an infinite graph that is false: one vertex that can be paired with anybody, and every other with exactly one, passes the test on every finite group and has no matching at all. The failure needs a vertex with infinitely many choices. Forbid that, and the theorem comes back, by an argument about trees rather than about matchings.
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Matching as they arrive
Applicants arrive one at a time, and each must be given a post or turned away on the spot. Any sensible rule matches at least half as many as the best assignment chosen with hindsight, and an adversary arranging the arrivals can hold every fixed rule to exactly half. Shuffle the posts once, at random, and give each arrival its best-ranked free post: the guarantee rises to 1 − 1/e, about 63%, and no rule of any kind can do better.
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Matching when the arrivals are shuffled
An adversary who chooses the order in which applicants arrive can hold any fixed matching rule to half the best. Take the order away and let it be random, and the plainest rule of all — give each arrival its first free place in a fixed list — rises to 1 − 1/e, because shuffling the arrivals turns out to be exactly the trick RANKING plays with the places, seen from the other side. Shuffle both and the guarantee rises again, to somewhere between 0.696 and 0.727, and where in that interval it lies is not known.
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Signs that make a determinant count
Hall's theorem says whether a board can be covered by dominoes. How many ways it can be covered is a different question, and for a general graph a hopeless one — the count is a permanent, and nobody expects an efficient way to compute permanents. On a flat board it is a determinant. Put a minus sign on the vertical pairs in every second column and the thirty-six coverings of a four-by-four board stop cancelling and add up; the chessboard has 12,988,816. A hole in the board breaks the rule in a way a cut repairs, and on K₃,₃, which cannot be drawn flat, no choice among its 512 signings works at all.
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The diamond that freezes its corners
A square board's domino tilings grow like 1.3385 choices per square. Arrange the same squares in a staircase-edged diamond and the count is exactly 2 to the n(n+1)/2 — only 1.1892 choices per square, at every size. The boundary has reached into the middle. Pick one of the diamond's tilings at random and the reason is visible: outside the circle inscribed in the diamond the dominoes are frozen into solid brickwork, and only inside it is the tiling free.