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.
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 1-page 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).
Electronic Notes in Discrete Mathematics only publishes short notes in discrete mathematics.
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.
Discrete Mathematics Letters http://www.dmlett.com/ publishes only short notes (that is, the papers consisting of maximum 8 published pages).