Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG(3,q)(3,q)

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective $3$-space PG$(3,q)$, with two vertices being adjacent if and only if the corresponding chambers are in general position. We show that a maximal independent set of vertices of $\Gamma$ contains $q^4+3q^3+4q^2+3q+1$, or $3q^3+5q^2+3q+1$, or at most $3q^3+4q^2+3q+2$ elements. For $q\geq 4$ the structure of the largest maximal independent sets is described. For $q\geq 7$ the structure of the maximal independent sets of the three largest cardinalities is described. Using the cardinality of the second largest maximal independent sets, we show that the chromatic number of $\Gamma$ is $q^2+q$

Calibration of the CMS hadron calorimeters using proton-proton collision data at root s=13 TeV

Methods are presented for calibrating the hadron calorimeter system of theCMSetector at the LHC. The hadron calorimeters of the CMS experiment are sampling calorimeters of brass and scintillator, and are in the form of one central detector and two endcaps. These calorimeters cover pseudorapidities vertical bar eta vertical bar ee data. The energy scale of the outer calorimeters has been determined with test beam data and is confirmed through data with high transverse momentum jets. In this paper, we present the details of the calibration methods and accuracy.Peer reviewe