We prove that every set system of bounded VC-dimension has a fractional Helly property. More precisely, if the dual shatter function of a set system F is bounded by o(m k), then F has fractional Helly number k. This means that for every > 0 there exists a > 0 such that if F 1 ; F 2 ; : : : ; F n 2 F are sets with T i2I F i 6 = ; for at least ? n k sets I f1; 2; : : :; ng of size k, then there e...

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward’s decomposition theorem for weakly chordal graphs, relying on a Roussel–Rubio-type lemma. We recall how Roussel–Rubio-type lemmas yield very short proofs of the existence of even pairs ...

We show that every cubic bridgeless graph with n vertices has at least 3n/4 − 10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.

We will extend Reed's Semi-Strong Perfect Graph Theorem by proving that unbreakable C 5-free graphs diierent from a C 6 and its complement have unique P 4-structure.

In 1891, Peterson [6] proved that every 3-regular bridge-less graph has a perfect matching. It is well known that the dual of a triangular mesh on a compact manifold is a 3-regular graph. M. Gopi and D. Eppstein [4] use Peterson’s theorem to solve the problem of constructing strips of triangles from triangular meshes on a compact manifold. P. Diaz-Gutierrez and M. Gopi [3] elaborate on the crea...

A dimension invariance property of finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation ...

Many examples of gravitational duals exist of theories that are highly supersymmetric and conformal in the UV yet have the same massless states as N = 2, 1, 0 QCD. We discuss such theories with an explicit UV cutoff and propose that, by tuning higher dimension operators at the cutoff by hand, the effects of the extra matter states in the UV may be removed from the IR physics. We explicitly work...

The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.