Uniformly random $k^2$-vertex graphs have clique size $O(\log k)$, well under $k$, and independent set size also $O(\log k)$, implying that their chromatic number is $\Omega(k^2/\log k)$. As for $k^2$-vertex triangle-free graphs, their chromatic number can be $\Theta(k/\sqrt{\log k})$ (and not higher); see Kim, Jeong Han (1995), "The Ramsey number $R(3,t)$ ...


Compactness for FOL was done in HOL by John Harrison, and reported at TPHOLs 1998. See Formalizing basic first order model theory.


If LOGLOG = NLOGLOG then LOG = NLOG. See more in: https://www.sciencedirect.com/science/article/pii/0304397590900086 and therefore, your question is still an unsolved problem.

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