  We consider zero-error communication over a two-transmitter deterministic
adversarial multiple access channel (MAC) governed by an adversary who has
access to the transmissions of both senders (hence called omniscient) and aims
to maliciously corrupt the communication. None of the encoders, jammer and
decoder is allowed to randomize using private or public randomness. This
enforces a combinatorial nature of the problem. Our model covers a large family
of channels studied in the literature, including all deterministic discrete
memoryless noisy or noiseless MACs. In this work, given an arbitrary
two-transmitter deterministic omniscient adversarial MAC, we characterize when
the capacity region
  1) has nonempty interior (in particular, is two-dimensional);
  2) consists of two line segments (in particular, has empty interior);
  3) consists of one line segment (in particular, is one-dimensional);
  4) or only contains $ (0,0) $ (in particular, is zero-dimensional).
  This extends a recent result by Wang, Budkuley, Bogdanov and Jaggi (2019)
from the point-to-point setting to the multiple access setting. Indeed, our
converse arguments build upon their generalized Plotkin bound and involve
delicate case analysis. One of the technical challenges is to take care of both
"joint confusability" and "marginal confusability". In particular, the
treatment of marginal confusability does not follow from the point-to-point
results by Wang et al. Our achievability results follow from random coding with
expurgation.
