It is shown that Rota’s basis conjecture follows from a similar conjecture that involves only three bases instead of n bases. Two counterexamples to the analogous conjecture involving only two bases are presented. [Note added 18 May 2005: Colin McDiarmid found a counterxample to Conjecture 2 after reading a previous version of this paper. See the end of the paper for details.]

We show that Stanley{Wilf enumerative conjecture on permutations follows easily from F uredi{Hajnal extremal conjecture on 0-1 matrices. We apply the method of deriving an (almost) exponential bound on number of objects from a (almost) linear bound on their sizes, which was discovered by Alon and Friedgut. They proved by it a weaker form of Stanley{Wilf conjecture. Using bipartite graphs, we gi...

Using a new technique, we prove rich family of special cases the matroid intersection conjecture. Roughly, conjecture for pairs tame matroids which have common decomposition by 2-separations into finite parts.

We prove that vertex-reinforced random walk on Z with weight of order kα, for α ∈ [0, 1/2), is recurrent. This confirms a conjecture of Volkov for α < 1/2. The conjecture for α ∈ [1/2, 1) remains open.

In this article we define a minor relation, which is stronger than the classical one, but too strong to become a well-quasi-order on the class of finite graphs. Nevertheless, with this terminology we are able to introduce a conjecture, which would imply the Lovasz conjecture and give an interesting insight on the symmetry of vertex-transitive graphs, if true. Though it could become an approach ...

We present a conjecture which unifies several conjectures on motives in characteristic p. Introduction In a talk at the June 1996 Oberwolfach algebraic K-theory conference, I explained that, in view of Voevodsky’s proof of the Milnor conjecture, the Bass conjecture on finite generation of K-groups of regular schemes of finite type over the integers implies the rational Beilinson-Soulé conjectur...

We study the Log-rank Conjecture for XOR functions and the Sensitivity Conjecture by exploiting structure of the Fourier transform and polynomial representations of Boolean functions. For the Log-rank Conjecture for XOR functions, we show that the it is true for functions with small spectral norm or low F2-degree. More precisely, we prove the following two bounds. 1. CC(f(x⊕ y)) = O(‖f̂‖1 · log ...

At the 16th British Combinatorial Conference (1997), Cameron introduced a new concept called 2-simultaneous coloring. He used this concept to reformulate a conjecture of Keedwell (1994) on the existence of critical partial latin squares of a given type. Using computer programs, we have veri ed the truth of the above conjecture (the SE conjecture) for all graphs having less than 29 edges. In thi...

This paper introduces 4−moves, a class of tangle replacement moves called rational pq − moves. We demonstrate the strength of invariants of 4 −moves and make modification to some long standing 4 −moves problems like Nakanishi’s 4 − moves conjecture[11]. We present the results and examples to show that Nakanishi’s 4 −moves conjecture is true for all knots upto 12 crossings. We suggest that the l...

The Berge–Sauer conjecture (see [2, 3]) says that any simple (no multiple edges and loops) 4-regular graph contains a 3-regular subgraph. This conjecture was proved in [4, 6]. In [1, 2] the Chevalley–Warning theorem was used to extend this result to graphs with multiple edges, which are 4-regular plus an edge. Our main result, Theorem 2.2, presents the sufficient condition for a 4-regular graph...