Let ${\mathrm{ex}}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform complete graph on $r+1$ vertices).Is every $r$-hypergraph $G$ on $n$ vertices the union of at most ${\mathrm{ex}}_r(n;K_{r+1}^r)$ many copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Here “(K_r^r)” is just a **single $r$-edge**, so your condition “no two share a (K_r^r)” is exactly the requirement that the chosen copies of (K_r^r) and (K_{r+1}^r) are **edge-disjoint** [[nomath]](i.e. you are partitioning $E(G)$ into single edges and $(r+1)$-vertex $r$-uniform cliques)[[/nomath]].

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.