TL;DR
The discussion of space-bounded nondeterminism is discussed and the theorem that NSPACE is closed under complement is shown, followed by introducing polynomial-time hierarchy and proving some of its basic properties.
We prove that for every acyclic matrix pattern $P$ it holds that $\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$.
We show that there exists $C\in\mathbb{N}$ such that $K_t$-minor free graphs are $Ct$-colorable. The proof was found by GPT-6 Astra, following the directions by the authors.
We prove that a relation obtained by composing arbitrary functions of arity $\leq k$ with a relation definable in an $n$-dependent structure is $kn$-dependent. This confirms a conjecture of Chernikov and Hempel. We also demonstrate optimality of the result.
It is shown that the (scattered) subsequence problem for two words represented by straight-line programs is PSPACE-complete, even over a binary alphabet. The lower bound is obtained by a polynomial-time reduction from quantified subset sum.
Using ChatGPT 5.6, we find a counterexample to the stable forking conjecture. This answers a long-standing open question of Hart, Kim and Pillay (1996).
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