Related papers: Intersection of a partitional and a general infini…

In this paper, we provide a combinatorial characterization of those collections of cells whose inner $2$-minor ideals are complete intersections. More precisely, given a collection of cells $\mathcal C$ and its associated inner $2$-minor…

In the matroid intersection problem, we are given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ defined on the same ground set $V$ of $n$ elements, and the objective is to find a common…

We introduce the following generalization of set intersection via characteristic vectors: for $n,q,s, t \ge 1$ a family $\mathcal{F}\subseteq \{0,1,\dots,q\}^n$ of vectors is said to be \emph{$s$-sum $t$-intersecting} if for any distinct…

Let $M$ be a matroid. We study the expansions of $M$ mainly to see how the combinatorial properties of $M$ and its expansions are related to each other. It is shown that $M$ is a graphic, binary or a transversal matroid if and only if an…

For matroids M and N on disjoint sets S and T, a semidirect sum of M and N is a matroid K on the union of S and T that, like the direct sum and the free product, has the restriction of K to S equal to M and the contraction of K to T equal…

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. We say that a pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and…

The maximum intersection problem for a matroid and a greedoid, given by polynomial-time oracles, is shown $NP$-hard by expressing the satisfiability of boolean formulas in 3-conjunctive normal form as such an intersection. The corresponding…

Given two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ over the same ground set, the matroid intersection problem is to find the maximum cardinality common independent set. In the weighted version of the problem, the goal is to find a…

Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented…

Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is…

A circuit-cocircuit intersection, or a CCI for short, of a matroid is the intersection of a circuit and a cocircuit. Oxley conjectured (1992) that a matroid with a CCI of size $k\ge4$ has a CCI of size $k-2$. We show that the conjecture…

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang…

We show that the fact that a political decision filtered through a finite tree of committees gives a determined answer generalises in some sense to infinite trees. This implies a new special case of the Matroid Intersection Conjecture.

Matroid intersection is a classical optimization problem where, given two matroids over the same ground set, the goal is to find the largest common independent set. In this paper, we show that there exists a certain "sparsifer": a subset of…

This is essentially an erratum, with some example to indicate inconsistencies. Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$. The Complete Intersection conjecture states that, for any ideal $I$ in $A$,…

We generalize the 1/3-2/3 conjecture from partially ordered sets to antimatroids: we conjecture that any antimatroid has a pair of elements x,y such that x has probability between 1/3 and 2/3 of appearing earlier than y in a uniformly…

We prove the complete intersection theorem and complete nontrivial-intersection theorem for systems of set partitions

We remark that Pearl's Graphoid intersection property, also called intersection property in Bayesian networks, is a particular case of a general intersection property, in the sense of intersection of coverings, for factorisation spaces,…

The "finite intersection property" for bifunctions is introduced and its relationship with generalized monotonicity properties is studied. Some results concerning existence of solution for (quasi-)equilibrium problems are established and…

A result [The intersection of a matroid and a simplicial complex, Trans. Amer. Math. Soc. 358] from 2006 of Aharoni and the first author of this paper states that for any two positive integers $p,q$, where $p$ divides $q$, if a matroid…