AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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For a graph $H$, the $H$-freeness edge-deletion problem is the algorithmic problem of finding, for an input graph $G$, the minimum number of edges of $G$ whose deletion turns $G$ into an $H$-free graph. We show that for every graph $H$ containing a cycle, this problem is NP-hard. This proves a conjecture of Gishboliner...
We prove the following two upper bounds for ordered Ramsey numbers: (1) Every ordered forest $F$ on $n$ vertices satisfies $R_{<}(F,F)=O(n^{1+\lceil\log\chi_{<}(F)\rceil})$. This in particular answers a question of Geneson, Holmes, Liu, Neidinger, Pehova and Wass. (2) There is a function $f$ such that, for every fixed...
We prove that for every acyclic matrix pattern $P$ it holds that $\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$.
Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic g...
Many important results in extremal graph theory can be roughly summarized as “if a triangle‐free graph G$$ G $$ has certain properties, then it has a homomorphism to a triangle‐free graph Γ$$ \Gamma $$ of bounded size.” For example, bounds on homomorphism thresholds give such a statement if G$$ G $$ has sufficiently hi...
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