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

We prove a vertex domination conjecture of Erdős, Faudree, Gould, Gyárfás, Rousseau, and Schelp, that for every n-vertex complete graph with edges coloured using three colours there exists a set of at most three vertices which have at least 2n/3 neighbours in one of the colours. Our proof makes extensive use of the ideas presented in "A New Bound for the 2/3 Conjecture" by Král', Liu, Sereni, Whalen, and Yilma.

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We determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends a previous result of Bondy, Shen, Thomassé and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. To be precise we show that a suitably weighted copy of the graph formed by deleting a certain 9-cycle from K3,3,3 has minimal triangle density among all weighted tripartite graphs with prescribed edge densities.