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LARGE
$\mathcal {F}$-FREE SUBGRAPHS IN 
$r$-CHROMATIC GRAPHS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 2 / October 2023
- Published online by Cambridge University Press:
- 02 December 2022, pp. 200-204
- Print publication:
- October 2023
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- Article
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For a graph G and a family of graphs
$\mathcal {F}$, the Turán number 
${\mathrm {ex}}(G,\mathcal {F})$ is the maximum number of edges an 
$\mathcal {F}$-free subgraph of G can have. We prove that 
${\mathrm {ex}}(G,\mathcal {F})\ge {\mathrm {ex}}(K_r, \mathcal {F})$ if the chromatic number of G is r and 
$\mathcal {F}$ is a family of connected graphs. This result answers a question raised by Briggs and Cox [‘Inverting the Turán problem’, Discrete Math. 342(7) (2019), 1865–1884] about the inverse Turán number for all connected graphs.
