The Existence of Exactlym-Coloured Complete Subgraphs

AbstractGiven a graphG, its edges are said to be exactlyx-coloured if we have a surjective map from the edges to some set of colours of sizex. Erickson considered the following statement which he denotedP(c,m): if the edges ofKω—the complete graph on vertex set N—are exactlyc-coloured, then there exists an infinite complete subgraph ofKωwhose edges are exactlym-coloured. Ramsey's Theorem states thatP(c,m) is true form=1 and allc⩾1, and can easily be used to show thatP(c,m) holds whenm=2 andc⩾2. Erickson conjectured thatP(c,m) is false wheneverc>m⩾3. We prove that givenm⩾3 there exists an integerC(m) such thatP(c,m) is false for allc⩾C(m)

The Existence of Exactly m-coloured Complete Subgraphs

this paper we will describe our main new method for constructing counterexamples. This will enable us to prove the following result

2-Partition-Transitive Tournaments

Given a tournament score sequence s1 s2 sn , we prove that there exists a tournament T on vertex set f1; 2; : : : ; ng such that the degree of any vertex i is s i and the subtournaments of T on both the even and the odd vertices are transitive in the given order. This means that i beats j whenever i < j and i j (mod 2). For an

Shadows in Linear Lattices

this paper we investigate the Kruskal-Katona type problem for linear lattices. First (section 2) we show that there is a dual problem to the minimization problem, namely the maximization of the cardinality of the "reverse shadow" which will be defined later. We prove that the minimization problem and the maximization problem are equisolvable. In section 3 we relax the problem by asking for which subset S of (GF (q)

Laboratory Observations of Ultra-Low Frequency Analogue Waves Driven by the Right-Hand Resonant Ion Beam Instability.

The Right-Hand Resonant Instability (RHI) is one of several electromagnetic ion/ion beam instabilities responsible for the formation of parallel magnetized collisionless shocks and the generation of ultra-low frequency (ULF) waves in their foreshocks. This instability has been observed for the first time under foreshock-relevant conditions in the laboratory through the repeatable interaction of a preformed magnetized background plasma and a super-Alfvénic laser-produced plasma. This platform has enabled unprecedented volumetric measurements of waves generated by the RHI, revealing filamentary current structures in the transverse plane. These measurements are made in the plasma rest frame with both high spatial and temporal resolution, providing a perspective that is complementary to spacecraft observations. Direct comparison of data from both the experiment and the Wind spacecraft to 2D hybrid simulations demonstrates that the waves produced are analogous to the ULF waves observed upstream of the terrestrial bow shock

Positron emission tomography in the assessment of left ventricular function in healthy rats: A comparison of four imaging methods

Objective To measure left ventricular (LV) function parameters in heart of healthy rats by three different positron emission tomography (PET) imaging techniques and by magnetic resonance imaging (MRI). Methods ECG-gated microPET examinations were obtained in seven healthy rats with 2-deoxy-2-[18F]fluoro-d-glucose (FDG) for calculation of LV-function from the blood-pool phase of the dynamic recording (FDGBP), and also from the later myocardial uptake (FDGMyo). On subsequent days, we re-measured LV-function using the novel blood-pool tracer 68Ga-albumin (AlbBP) and again by FDG (FDGMyo2) in one setting. Cine-MRI examination provided the reference standard measurement. Results The mean LV ejection fractions (LVEF) were 56 ± 3 (FDGBP), 55 ± 3 (FDGMyo), 56 ± 3 (FDGMyo2), 57 ± 3 (AlbBP), and 57 ± 2 (MRI). There were good to excellent correlations found between the LVEF-values as compared to MRI reference standard for FDGBP (r = 0.71), FDGMyo (r = 0.86) and AlbBP (r = 0.88). Both of the blood-pool methods significantly overestimated the magnitudes of end-diastolic-volume and end-systolic-volume, whereas FDGMyo matched closely to the MRI reference standard. There was no significant bias for both blood-pool methods and a minor negative bias for FDGMyo regarding the LV ejection fraction (LVEF) when compared to cine-MRI results. There was no significant difference between the means of FDGMyo and FDGMyo2 (P = .50). Conclusions Relative to reference standard MRI measurements of LVEF, there was excellent agreement between PET-based measurements, notably for the novel blood-pool tracer 68Ga-albumin