
TL;DR
This paper investigates the probability that a random collection of k-subsets forms the bases of a matroid, analyzing phase transitions, structural properties, and implications for counting matroids and related combinatorial objects.
Contribution
It provides asymptotic probability results for random matroid bases, identifies conditions preventing matroid formation, and improves estimates for the number of matroids and sparse paving matroids.
Findings
Probability of random collection forming a matroid undergoes phase transition.
When a matroid occurs, it is typically a sparse paving matroid.
Improved estimates for the number of matroids, paving matroids, and sparse paving matroids.
Abstract
Let be a random collection of -subsets of where each possible set is present independently with probability . Let be the event that defines the set of bases of a matroid. We prove that If where , then \[ \lim_{n\to\infty}\Pr[\cal E_{\cal B}\mid |\cal B|\geq2]=\begin{cases}1&c_n\to0.\\e^{-c^2/2}&c_n\to c.\\0&c_n\to \infty.\end{cases}\] In addition, we identify a condition preventing the occurence of and prove a hitting time version for the occurence of . We also prove that when occurs, defines a sparse paving matroid w.h.p. In addition, study a greedy algorithm that produces a random matroid defined by a collection of hyperplanes. We use this to improve the estimates in…
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