TITLE: Examples of notably long or difficult proofs that only improve upon existing results by a small amount
QUESTION [106 upvotes]: I was recently reading Bui, Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros on the critical line of the Riemann $\zeta$-function. In order to achieve this, they define the (monstrous) mollifier:
$$
\psi=\sum_{n\leq y_1}\frac{\mu(n)P_1[n]n^{\sigma_0-\frac{1}{2}}}{n^s}+\chi(s+\frac{1}{2}-\sigma_0)\sum_{hk\leq y_2}\frac{\mu_2(h)h^{\sigma_0-\frac{1}{2}}k^{\frac{1}{2}-\sigma_0}}{h^sk^{1-s}}P_2(hk).
$$
This is followed by a nearly 20-page tour-de-force of grueling analytic number theory, coming to a grinding halt by improving the already known bounds from $40.88\%$ to... $41.05\%$.
Of course any improvement of this lower bound is a highly important achievement, and the methods and techniques used in the paper are deeply inspirational. However, given the highly complex nature of the proof and the small margin of improvement obtained over already existing results, I was inspired to ask the following question: 
What are other examples of notably long or difficult proofs that only improve upon existing results by a small amount?

REPLY [7 votes]: The following example is described in The Man Who Loved Only Numbers by Paul Hoffman. 
There is a reasonably short proof, found by Esther Klein (later Szekeres) in 1932, that given 5 points in the plane, there exist 4 of them that form a convex quadrilateral. The minimum number of points needed such that there always exists 5-element subset forming a convex pentagon is 9. There is a conjecture that the minimum number of points needed such that there exists an $n$-element subset forming a convex $n$-gon is $2^{n-2} + 1$. It is known it has to be at least that. This is called the "Happy Ending Problem" (apparently because George Szekeres had impressed Esther so much with his proof that there is a minimum finite number for each $n$ that he won her hand in marriage). 
For the case $n = 6$, the best that Erdős achieved was 71 points, a somewhat distant cry from the conjectured 17. This was sometime in the 1930's. After the memorial service for Erdős in 1996, Ronald Graham and his wife Fan Chung thought it was high time that someone have another crack at it, after 60 years -- and were very excited that during a long flight to New Zealand, they managed to lower it down by just a single point, to 70. This required introducing new ideas. 
(As Graham explains, Kleitman and Pachner soon lowered it further to 65. There were subsequent further improvements until finally George Szekeres and his collaborator Peters reached 17, with the help of computers, in a paper published in 2006.)