English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

The critical threshold of a (simple binary) matroid $N$ is the infimum overall $\rho$ such that any $N$-free matroid $M$ with $|M|>\rho2^{r(M)}$ hasbounded critical number. In this paper, we resolve two conjectures of Geelenand Nelson, showing that the critical threshold of the projective geometry$PG(t-1,2)$ is $1-3\cdot2^{-t}$. We do so by proving the following strongerstatement: if $M$ is $PG(t-1,2)$-free with $|M|>(1-3\cdot2^{-t})2^{r(M)}$, thenthe critical number of $M$ is $t-1$ or $t$. Together with earlier results ofGeelen and Nelson [GN14] and Govaerts and Storme [GS06], this completes theclassification of dense $PG(t-1,2)$-free matroids.

Dense binary $PG(t-1,2)$-free matroids have critical number $t-1$ or $t$