It's pretty much common knowledge that all things Gygax(TM) are, well, trademarked by The Trustees of the GAIL C. GYGAX REVOCABLE TRUST (what's a "revocable trust" you may ask - that, my friends, is time for another post).
Gygaxian has an application for trademark (from the same The Trustees of the GAIL C. GYGAX REVOCABLE TRUST) with a letter of suspension. Which means, no, they don't have the trademark yet BUT they have reserved their first place in line IF its decided the word is trademark-able.
Interestingly enough, one must use Gygax(TM) itself as an adjective, which would seem to make the trademarking of Gygaxian a bit of a reach, as it isn't in and of itself a product or a brand. Below explains the proper use of Gygax(TM) when referring to the brand and not the man:
Now, of course, there hasn't been a new Gygax RPG product released by the Trust in, like, ever (or never). So using Gygax(TM) as an adjective isn't going to come up much. Unless you refer to D&D as a Gygax RPG,
Back to Gygaxian. Its been in common usage in the RPG community AT LEAST since 2007, and that's where the trademark would apply - with RPGs.
https://forum.rpg.net/showthread.php?361362-Gygaxian-Dungeon October, 2007
http://www.engadget.com/2008/03/13/off-the-grid-gygaxian-game-design/ March, 2008
http://grognardia.blogspot.com/2008/09/gygaxian-naturalism.html September, 2008
http://paizo.com/threads/rzs2iuh6?4Es-Rejection-of-Gygaxian-Naturalism October, 2008
http://www.urbandictionary.com/define.php?term=gygaxian January, 2009
http://dnd.wizards.com/articles/features/waxing-gygaxian March, 2012
https://www.reddit.com/r/rpg/comments/2f8blk/serious_why_is_gygaxian_dd_considered_antiplayer/ September, 2014
edit - wow, hat tip to Davrion - mentions here go back to the 90s!
(there are hundreds more uses to be found with a nice Google search)
How does the Urban Dictionary define Gygaxian?
If a word is already in common usage, it is harder to claim a trademark on it.
Want to keep Gygaxian free from trademarking? Use it freely and use it well. We still might lose it, but without usage, the odds shift in an ill direction.