On the number of matroids

We consider the problem of determining $m_n$, the number of matroids on $n$ elements. The best known lower bound on $m_n$ is due to Knuth (1974) who showed that $\log \log m_n$ is at least $n-3/2\log n-1$. On the other hand, Piff (1973) showed that $\log\log m_n\leq n-\log n+\log\log n +O(1)$, and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on $\log\log m_n$ that is within an additive $1+o(1)$ term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.Comment: Final version, 17 page

Wide partitions, Latin tableaux, and Rota's basis conjecture

Say that mu is a ``subpartition'' of an integer partition lambda if the multiset of parts of mu is a submultiset of the parts of lambda, and define an integer partition lambda to be ``wide'' if for every subpartition mu of lambda, mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of mu). Then Brian Taylor and the first author have conjectured that an integer partition lambda is wide if and only if there exists a tableau of shape lambda such that (1) for all i, the entries in the ith row of the tableau are precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the entries in the jth column of the tableau are pairwise distinct. This conjecture was originally motivated by Rota's basis conjecture and, if true, yields a new class of integer multiflow problems that satisfy max-flow min-cut and integrality. Wide partitions also yield a class of graphs that satisfy ``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above conjecture implies that these graphs furthermore have a completely saturated stable set partition. We present several partial results, but the conjecture remains very much open.Comment: Joined forces with Goemans and Vondrak---several new partial results; 28 pages, submitted to Adv. Appl. Mat

Clique versus Independent Set

Yannakakis' Clique versus Independent Set problem (CL-IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size $O(n^{\log n})$, and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a polynomial CS-separator almost surely exists for random graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a clique and a stable set) then there exists a constant $c_H$ for which we find a $O(n^{c_H})$ CS-separator on the class of H-free graphs. This generalizes a result of Yannakakis on comparability graphs. We also provide a $O(n^{c_k})$ CS-separator on the class of graphs without induced path of length k and its complement. Observe that on one side, $c_H$ is of order $O(|H| \log |H|)$ resulting from Vapnik-Chervonenkis dimension, and on the other side, $c_k$ is exponential. One of the main reason why Yannakakis' CL-IS problem is fascinating is that it admits equivalent formulations. Our main result in this respect is to show that a polynomial CS-separator is equivalent to the polynomial Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition into k complete bipartite graphs, then its chromatic number is polynomially bounded in terms of k. We also show that the classical approach to the stubborn problem (arising in CSP) which consists in covering the set of all solutions by $O(n^{\log n})$ instances of 2-SAT is again equivalent to the existence of a polynomial CS-separator

Problems on Matchings and Independent Sets of a Graph

Let $G$ be a finite simple graph. For $X \subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\max \, \{d(X):X\subset V(G)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with $d(S) > 0$ if no proper subset of $S$ has positive difference. A graph $G$ is called K\"onig-Egerv\'ary if the sum of its independence number ($\alpha (G)$) and matching number ($\mu (G)$) equals $|V(G)|$. It is known that bipartite graphs are K\"onig-Egerv\'ary. In this paper, we study independent sets with positive difference for which every proper subset has a smaller difference and prove a result conjectured by Levit and Mandrescu in 2013. The conjecture states that for any graph, the number of inclusion minimal sets $S$ with $d(S) > 0$ is at least the critical difference of the graph. We also give a short proof of the inequality $|$ker$(G)| + |$diadem$(G)| \le 2\alpha (G)$ (proved by Short in 2016). A characterization of unicyclic non-K\"onig-Egerv\'ary graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha (G) - \mu (G)$, is proved. We also make an observation about ker$G)$ using Edmonds-Gallai Structure Theorem as a concluding remark.Comment: 18 pages, 2 figure

Covering graphs by monochromatic trees and Helly-type results for hypergraphs

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph $H$ if it is known that any collection of a few edges of $H$ has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao, Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi