We say that α ∈ [0, 1) is a jump for an integer r 2 if there exists c(α) > 0 such that for all > 0 and all t 1, any r-graph with n n 0 (α, , t) vertices and density at least α + contains a subgraph on t vertices of density at least α + c.The Erdős-Stone-Simonovits theorem [4,5] implies that for r = 2, every α ∈ [0, 1) is a jump. Erdős [3] showed that for all r 3, every α ∈ [0, r!/r r ) is a jump. Moreover he made his famous 'jumping constant conjecture', that for all r 3, every α ∈ [0, 1) is a jump. Frankl and Rödl [7] disproved this conjecture by giving a sequence of values of non-jumps for all r 3.We use Razborov's flag algebra method [9] to show that jumps exist for r = 3 in the interval [2/9, 1). These are the first examples of jumps for any r 3 in the interval [r!/r r , 1). To be precise, we show that for r = 3 every α ∈ [0.2299, 0.2316) is a jump.We also give an improved upper bound for the Turán density of K − 4 = {123, 124, 134}: π(K − 4 ) 0.2871. This in turn implies that for r = 3 every α ∈ [0.2871, 8/27) is a jump.

If F is a family of graphs then the Turán density of F is determined by the minimum chromatic number of the members of F.The situation for Turán densities of 3-graphs is far more complex and still very unclear. Our aim in this paper is to present new exact Turán densities for individual and finite families of 3-graphs, in many cases we are also able to give corresponding stability results. As well as providing new examples of individual 3-graphs with Turán densities equal to 2/9, 4/9, 5/9 and 3/4 we also give examples of irrational Turán densities for finite families of 3-graphs, disproving a conjecture of Chung and Graham. (Pikhurko has independently disproved this conjecture by a very different method.)A central question in this area, known as Turán's problem, is to determine the Turán density of K (3) 4

How large can the Lagrangian of an r-graph with m edges be? Frankl and Füredi [1] conjectured that the r-graph of size m formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs of size m. We prove the first ‘interesting’ case of this conjecture, namely that the 3-graph with (t3) edges and largest Lagrangian is [t](3). We also prove that this conjecture is true for 3-graphs of several other sizes.For general r-graphs we prove a weaker result: for t sufficiently large, the r-graph of size (tr) supported on t + 1 vertices and with largest Lagrangian, is [t](r).

Context. Disk-integrated photometric data of asteroids do not contain accurate information on shape details or size scale. Additional data such as disk-resolved images or stellar occultation measurements further constrain asteroid shapes and allow size estimates. Aims. We aim to use all available disk-resolved images of about forty asteroids obtained by the Near-InfraRed Camera (Nirc2) mounted on the W.M. Keck II telescope together with the disk-integrated photometry and stellar occultation measurements to determine their volumes. We can then use the volume, in combination with the known mass, to derive the bulk density. Methods. We download and process all asteroid disk-resolved images obtained by the Nirc2 that are available in the Keck Observatory Archive (KOA). We combine optical disk-integrated data and stellar occultation profiles with the disk-resolved images and use the All-Data Asteroid Modeling (ADAM) algorithm for the shape and size modeling. Our approach provides constraints on the expected uncertainty in the volume and size as well.Results. We present shape models and volume for 41 asteroids. For 35 asteroids, the knowledge of their mass estimates from the literature allowed us to derive their bulk densities. We clearly see a trend of lower bulk densities for primitive objects (C-complex) than for S-complex asteroids. The range of densities in the X-complex is large, suggesting various compositions. Moreover, we identified a few objects with rather peculiar bulk densities, which is likely a hint of their poor mass estimates. Asteroid masses determined from the Gaia astrometric observations should further refine most of the density estimates.

For a graph G and integer r ≥ 1 we denote the collection of independent r-sets of G by I (r) (G). If v ∈ V (G) then I (r) v (G) is the collection of all independent r-sets containing v. A graph G, is said to be r-EKR, for r ≥ 1, iff no intersecting family A ⊆ I (r) (G) is larger than max v∈V (G) |I (r) v (G)|. There are various graphs which are known to have his property: the empty graph of order n ≥ 2r (this is the celebrated Erdős-Ko-Rado theorem), any disjoint union of at least r copies of K t for t ≥ 2, and any cycle. In this paper we show how these results can be extended to other classes of graphs via a compression proof technique.In particular we show that any disjoint union of at least r complete graphs, each of order at least two, is r-EKR. We also show that paths are r-EKR for all r ≥ 1.