An Exponential Value-Oracle Lower Bound for \({k}\)-Submodular Function Minimization

Proof of the Pach-Tardos conjecture

We prove that for every acyclic matrix pattern $P$ it holds that $\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$.

New Lower Bounds for Probabilistic Degree and AC0 with Parity Gates

Lecture 7: The Polynomial-Time Hierarchy

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.

A Proof of the Linear Hadwiger Conjecture

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.

Dictators are most informative

We prove the Courtade-Kumar conjecture: among all Boolean functions $f\colon \{-1,1\}^n\to\{-1,1\}$, a dictator retains the most information about a uniformly random input observed through independent binary noise.

The Matroid Secretary Conjecture is True

We resolve the matroid secretary conjecture, giving an online algorithm that accepts each element of the offline optimum with probability at least $1/4$. The algorithm only needs the number of elements in advance and independence-oracle access to subsets of already-arrived elements; it does not need to know the matroid...