MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

No eleventh conditional Ingleton inequality

Tobias Boege


A rational probability distribution on four binary random variables $X, Y, Z, U$ is constructed which satisfies the conditional independence relations $[X \perp\!\!\!\perp Y]$, $[X \perp\!\!\!\perp Z \mid U]$, $[Y \perp\!\!\!\perp U \mid Z]$ and $[Z \perp\!\!\!\perp U \mid XY]$ but whose entropy vector violates the Ingleton inequality.

This settles a recent question of Studený (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold.


Related publications

2023 Journal Open Access
Tobias Boege

No eleventh conditional Ingleton inequality

In: Experimental mathematics, (2023)