An extension of Tur\'an's Theorem, uniqueness and stability

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's comments incorporate

A density Corr\'adi-Hajnal Theorem

We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi-Hajnal Theorem.Comment: 41 pages (including 11 pages of appendix), 4 figures, 2 table

An extension of Turán's theorem, uniqueness and stability

We determine the maximum number of edges of an n -vertex graph G with the property that none of its r -cliques intersects a fixed set M⊂V(G) . For (r−1)|M|≥n , the (r−1) -partite Turán graph turns out to be the unique extremal graph. For (r−1)|M|<n , there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results

The tree packing conjecture for trees of almost linear maximum degree

We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result actually allows to replace the host graph $K_n$ by an arbitrary quasirandom graph, and to generalize from trees to graphs of bounded degeneracy that are rich in bare paths, contain some odd degree vertices, and only satisfy much less stringent restrictions on their number of vertices.Comment: 150 pages, 4 figure

Packing degenerate graphs

Given D and γ>0, whenever c>0 is sufficiently small and n sufficiently large, if G is a family of D-degenerate graphs of individual orders at most n, maximum degrees at most cnlogn, and total number of edges at most (1−γ)(n2), then G packs into the complete graph Kn. Our proof proceeds by analysing a natural random greedy packing algorithm

Genetic and Evolutionary Analyses of the Human Bone Morphogenetic Protein Receptor 2 (BMPR2) in the Pathophysiology of Obesity

Human bone morphogenetic protein receptor 2 (BMPR2) is essential for BMP signalling and may be involved in the regulation of adipogenesis. The BMPR2 locus has been suggested as target of recent selection in human populations. We hypothesized that BMPR2 might have a role in the pathophysiology of obesity.Evolutionary analyses (dN/dS, Fst, iHS) were conducted in vertebrates and human populations. BMPR2 mRNA expression was measured in 190 paired samples of visceral and subcutaneous adipose tissue. The gene was sequenced in 48 DNA samples. Nine representative single nucleotide polymorphisms (SNPs) were genotyped for subsequent association studies on quantitative traits related to obesity in 1830 German Caucasians. An independent cohort of 925 Sorbs was used for replication. Finally, relation of genotypes to mRNA in fat was examined.The evolutionary analyses indicated signatures of selection on the BMPR2 locus. BMPR2 mRNA expression was significantly increased both in visceral and subcutaneous adipose tissue of 37 overweight (BMI>25 and <30 kg/m²) and 80 obese (BMI>30 kg/m²) compared with 44 lean subjects (BMI< 25 kg/m²) (P<0.001). In a case-control study including lean and obese subjects, two intronic SNPs (rs6717924, rs13426118) were associated with obesity (adjusted P<0.05). Combined analyses including the initial cohort and the Sorbs confirmed a consistent effect for rs6717924 (combined P = 0.01) on obesity. Moreover, rs6717924 was associated with higher BMPR2 mRNA expression in visceral adipose tissue.Combined BMPR2 genotype-phenotype-mRNA expression data as well as evolutionary aspects suggest a role of BMPR2 in the pathophysiology of obesity