The journal Discrete Mathematics contains a lot of short notes (i.e., less than 7 journal pages). What are some other journals that publish short notes in discrete mathematics? I've looked at other journals, but most of them seem to contain primarily long papers.

1$\begingroup$ Which combinatorics journals have a policy against short papers? None that I know of. The question is more about the significance of the results, not the number of pages. $\endgroup$ – Brendan McKay Oct 11 '15 at 7:26
There are various general journals that are focused on "short papers." There a seven pages paper would fit just fine.
Examples include, Proceedings of the AMS, Bulletin of the LMS, Archiv der Mathematik. For more you could read Which journals publish 1page papers
Yet, as discussed there even most other journals do not have an actual lower bound on the length of the paper; of course the paper has to be sufficiently substantive.
Furthermore, it is not that rare for a discrete mathematics paper to be short. For most journal specialized in combinatorics and discrete mathematics a paper of that length should not be anything unusual (consult the instructions for authors though).
For an overview about discrete mathematics journals see for example Two questions about combinatorics journals or the respective answers on Top specialized journals
Electronic Notes in Discrete Mathematics only publishes short notes in discrete mathematics.

3$\begingroup$ But it looks like it doesn't allow direct submissions  it is only a venue for publications from conferences. $\endgroup$ – Alexi Oct 10 '15 at 21:06
The Electronic Journal of Combinatorics Current issue (issue 3, 2015), has the first two listed articles with $7$ and $8$ pages, so I assume there is no page minimum requirement.

2$\begingroup$ Correct. We used to distinguish "notes" from "articles" if they were 4 pages or less, but we don't distinguish them any more. $\endgroup$ – Brendan McKay Oct 10 '15 at 23:08
Also pay your attention to Algebra and Discrete Mathematics and Diskretnaya Matematika.
Here is an example. Here is another one.
Discrete Mathematics Letters http://www.dmlett.com/ publishes only short notes (that is, the papers consisting of maximum 8 published pages).