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# Dabeen Lee (이다빈), From coordinate subspaces over finite fields to ideal multipartite uniform clutters

## Tuesday, August 29, 2023 @ 4:30 PM - 5:30 PM KST

Take a prime power $q$, an integer $n\geq 2$, and a coordinate subspace $S\subseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$ an $n$-partite $n$-uniform clutter $\mathcal{C}$, where every part has size $q$ and there is a bijection between the vectors in $S$ and the members of $\mathcal{C}$. In this paper, we determine when the clutter $\mathcal{C}$ is *ideal*, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether $q$ is $2,4$, a higher power of $2$, or otherwise. Each characterization uses crucially that idealness is a *minor-closed property*: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of $\mathcal{C}$ depends solely on the underlying matroid of $S$. Our theorems also extend from idealness to the stronger *max-flow min-cut* property. As a consequence, we prove the Replication and $\tau=2$ conjectures for this class of clutters. This is joint work with Ahmad Abdi (London School of Economics).