On two problems in graph Ramsey theory

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi and Trotter states that there exists a constant c(\Delta) such that r(H) \leq c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The important open question is to determine the constant c(\Delta). The best results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta \log^2 \Delta}. We improve this upper bound, showing that there is a constant c for which c(\Delta) \leq 2^{c \Delta \log \Delta}. The induced Ramsey number r_{ind}(H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erd\H{o}s conjectured the existence of a constant c such that, for any graph H on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in the exponent.Comment: 18 page

Improving manual oxygen titration in preterm infants by training and guideline implementation

To study oxygen saturation (SpO2) targeting before and after training and guideline implementation of manual oxygen titration, two cohorts of preterm infants 21%. ABCs where oxygen therapy was given were identified and analyzed. After training and guideline implementation the %SpO2-wtr increased (median interquartile range (IQR)) 48.0 (19.6-63.9) % vs 61.9 (48.5-72.3) %; p 95% (44.0 (27.8-66.2) % vs 30.8 (22.6-44.5) %; p 95% did not decrease (73% vs 64%; ns) but lasted shorter (2 (0-7) vs 1 (1-3) minute; p < 0.004). CONCLUSION: Training and guideline implementation in manual oxygen titration improved SpO2 targeting in preterm infants with more time spent within the target range and less frequent hyperoxaemia. The durations of hypoxaemia and hyperoxaemia during ABCs were shorter. What is Known: • Oxygen saturation targeting in preterm infants can be challenging and the compliance is low when oxygen is titrated manually. • Hyperoxaemia often occurs after oxygen therapy for oxygen desaturation during apnoeas. What is New: • Training and implementing guidelines improved oxygen saturation targeting and reduced hyperoxaemia. • Training and implementing guidelines improved manual oxygen titration during ABC