论文标题
关于弗里克和贾法里的两个猜想
Concerning Two Conjectures of Frick and Jafari
论文作者
论文摘要
2022年，Hamid Reza Daneshpajouh对以下弗洛里安·弗里克（Florian Frick）的猜想提供了一些反例。 \ bf猜想。令$ r \ geq 3 $。然后，地面设置$ [n] $满足$$χ\ left（{\ rm kg}^r（{{\ cal g} _ {rm kg} _ {r- {rm {\ rm stable}}}） g}）} {r-1} \ right \ rceil。 $$在本文中，我们改善了Danashpajouh的结果。另外，我们为阿米尔·贾法里（Amir Jafari）的以下猜想提供了几个反例。 推测。如果$ s \ geq r \ geq 2 $，那么对于任何超graph $ {\ cal g} $，带有顶点set $ [n] $，我们有$$χ\ left（{\ rm kg}^r（{\ cal G} \ frac {{\ rm ecd}^s（{\ cal g}）} {r-1} \ right \rceil。$$
In 2022, Hamid Reza Daneshpajouh provided some counterexamples to the following conjecture of Florian Frick. \bf Conjecture. Let $r \geq 3$. Then, every hypergraph ${\cal G}$ over the ground set $[n]$ satisfies $$ χ\left({\rm KG}^r ({\cal G}_{r-{\rm stable}})\right) \geq \left\lceil \frac{{\rm cd}^r ({\cal G})}{r-1} \right\rceil . $$ In this paper, we improve Danashpajouh's results. Also, we provide several counterexamples to the following recent conjecture of Amir Jafari. Conjecture. If $s \geq r \geq 2$, then for any hypergraph ${\cal G}$ with vertex set $[n]$ we have $$ χ\left({\rm KG}^r ({\cal G}_{s-{\rm stable}})\right) \geq \left\lceil \frac{{\rm ecd}^s ({\cal G})}{r-1} \right\rceil .$$
