A celebrated unresolved conjecture of Erdős and Hajnal states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or stable set of size at least n (H). The conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H-free n...

LetG be a connected simple graph on n vertices. Gallai’s conjecture asserts that the edges ofG can be decomposed into n2 paths. Let H be the subgraph induced by the vertices of even degree in G. Lovász showed that the conjecture is true ifH contains at most one vertex. Extending Lovász’s result, Pyber proved that the conjecture is true if H is a forest. A forest can be regarded as a graph in wh...

In 1947, Lehmer conjectured that the Ramanujan τ -function τ(m) never vanishes for all positive integers m, where the τ(m) are the Fourier coefficients of the cusp form ∆24 of weight 12. Lehmer verified the conjecture in 1947 for m < 214928639999. In 1973, Serre verified up to m < 10, and in 1999, Jordan and Kelly for m < 22689242781695999. The theory of spherical t-design, and in particular th...

We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation...

We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation...

The h-principle is a concept introduced by Gromov which pertains to various problems in differential geometry, where one expects high flexibility of the moduli spaces of solutions due to the high-dimensionality (or underdetermined nature) of the problem. Interestingly, in some cases a form of the h-principle holds even for systems of partial differential relations which are, formally, not under...

A conjecture of Moore claims that if Γ is a group and H a finite index subgroup of Γ such that Γ −H has no elements of prime order (e.g. Γ is torsion free), then a Γ-module which is projective over H is projective over Γ. The conjecture is known for finite groups. In that case, it is a direct consequence of Chouinard’s theorem which is based on a fundamental result of Serre on the vanishing of ...

Let τ(H) be the cover number and ν(H) be the matching number of a hypergraph H. Ryser conjectured that every r-partite hypergraph H satisfies the inequality τ(H) ≤ (r − 1)ν(H). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with ν(H) = 1, Ryser’s conjecture reduces to τ(H) ≤ r − 1. Even this conjecture is extremely difficult and is open for all r ≥ 6. For infi...

The acyclic edge colouring problem is extensively studied in graph theory. The corner-stone of this field is a conjecture of Alon et. al.[1] that a′(G) ≤ ∆(G) + 2. In that and subsequent work, a′(G) is typically bounded in terms of ∆(G). Motivated by this we introduce a term gap(G) defined as gap(G) = a′(G) − ∆(G). Alon’s conjecture can be rephrased as gap(G) ≤ 2 for all graphs G. In [5] it was...

Associated with every graph G of chromatic number χ is another graph G. The vertex set of G consists of all χ-colorings of G, and two χ-colorings are adjacent when they differ on exactly one vertex. According to a conjecture of Björner and Lovász, this graph G must be disconnected. In this note we give a counterexample to this conjecture. One of the most disturbing problems in graph theory is t...