Problems in extremal graph theory

We consider a variety of problems in extremal graph and set theory. The {\em chromatic number} of $G$, $\chi(G)$, is the smallest integer $k$ such that $G$ is $k$-colorable. The {\it square} of $G$, written $G^2$, is the supergraph of $G$ in which also vertices within distance 2 of each other in $G$ are adjacent. A graph $H$ is a {\it minor} of $G$ if $H$ can be obtained from a subgraph of $G$ by contracting edges. We show that the upper bound for $\chi(G^2)$ conjectured by Wegner (1977) for planar graphs holds when $G$ is a $K_4$-minor-free graph. We also show that $\chi(G^2)$ is equal to the bound only when $G^2$ contains a complete graph of that order. One of the central problems of extremal hypergraph theory is finding the maximum number of edges in a hypergraph that does not contain a specific forbidden structure. We consider as a forbidden structure a fixed number of members that have empty common intersection as well as small union. We obtain a sharp upper bound on the size of uniform hypergraphs that do not contain this structure, when the number of vertices is sufficiently large. Our result is strong enough to imply the same sharp upper bound for several other interesting forbidden structures such as the so-called strong simplices and clusters. The {\em $n$-dimensional hypercube}, $Q_n$, is the graph whose vertex set is $\{0,1\}^n$ and whose edge set consists of the vertex pairs differing in exactly one coordinate. The generalized Tur\'an problem asks for the maximum number of edges in a subgraph of a graph $G$ that does not contain a forbidden subgraph $H$. We consider the Tur\'an problem where $G$ is $Q_n$ and $H$ is a cycle of length $4k+2$ with $k\geq 3$. Confirming a conjecture of Erd{\H o}s (1984), we show that the ratio of the size of such a subgraph of $Q_n$ over the number of edges of $Q_n$ is $o(1)$, i.e. in the limit this ratio approaches 0 as $n$ approaches infinity

Coloring Grids Avoiding Bicolored Paths

The vertex-coloring problem on graphs avoiding bicolored members of a family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of graphs (Gr\"unbaum, 1973) where bicolored copies of $P_4$ and cycles are not allowed, respectively. In this paper, we study a variation of this problem, by considering vertex coloring on grids forbidding bicolored paths. We let $P_k$-chromatic number of a graph be the minimum number of colors needed to color the vertex set properly avoiding a bicolored $P_k.$ We show that in any 3-coloring of the cartesian product of paths, $P_{k-2}\square P_{k-2}$, there is a bicolored $P_k.$ With our result, the problem of finding the $P_k$-chromatic number of product of two paths (2-dimensional grid) is settled for all $k.

Salt restriction in kidney disease—a missed therapeutic opportunity?

The importance of salt restriction in the treatment of patients with renal disease has remained highly controversial. In the following we marshal the current evidence that salt plays a definite role in the genesis of hypertension and target organ damage, point to practical problems of salt restriction, and report on novel pathomechanisms of how salt affects blood pressure and causes target organ damage

Hypertension in children with chronic kidney disease: pathophysiology and management

Arterial hypertension is very common in children with all stages of chronic kidney disease (CKD). While fluid overload and activation of the renin–angiotensin system have long been recognized as crucial pathophysiological pathways, sympathetic hyperactivation, endothelial dysfunction and chronic hyperparathyroidism have more recently been identified as important factors contributing to CKD-associated hypertension. Moreover, several drugs commonly administered in CKD, such as erythropoietin, glucocorticoids and cyclosporine A, independently raise blood pressure in a dose-dependent fashion. Because of the deleterious consequences of hypertension on the progression of renal disease and cardiovascular outcomes, an active screening approach should be adapted in patients with all stages of CKD. Before one starts antihypertensive treatment, non-pharmacological options should be explored. In hemodialysis patients a low salt diet, low dialysate sodium and stricter dialysis towards dry weight can often achieve adequate blood pressure control. Angiotensin-converting enzyme (ACE) inhibitors and angiotensin receptor blockers are first-line therapy for patients with proteinuria, due to their additional anti-proteinuric properties. Diuretics are a useful alternative for non-proteinuric patients or as an add-on to renin–angiotensin system blockade. Multiple drug therapy is often needed to maintain blood pressure below the 90th percentile target, but adequate blood pressure control is essential for better renal and cardiovascular long-term outcomes

Multiple-frequency bioimpedance devices for fluid management in people with chronic kidney disease receiving dialysis : a systematic review and economic evaluation

The National Institute for Health Research Health Technology Assessment programme. Acknowledgements The authors are grateful to Lara Kemp for her secretarial support. The authors would also like to thank the members of the specialist committee assembled to support this assessment: Dr Andrew Davenport (Royal Free Hospital, London), Dr Simon Roe (Nottingham University Hospitals NHS Trust), Dr Elizabeth Lindley (St James’s University Hospital), Dr Wesley Hayes (Great Ormond Street Hospital), Ms Joanne Prince (Central Manchester University Hospitals NHS Foundation Trust), Mr Nick McAleer (Royal Devon & Exeter NHS Foundation Trust), Dr Kay Tyerman (Leeds General Infirmary), Dr Graham Woodrow (St James’s University Hospital) and Mr Paul Taylor (lay specialist committee member). The Health Services Research Unit, Health Economics Research Unit and Institute of Applied Health Sciences, University of Aberdeen are all core funded by the Chief Scientist Office of the Scottish Government Health DirectoratesPeer reviewedPublisher PD

Pharmacological and non-pharmacological treatment of hypertension in dialysis patients

WOS: 000327884500020Hypertension is very common in dialysis patients. The most important cause of hypertension is hypervolemia. Fluid restriction and volume management with standard hemodialysis are effective strategies to achieve dry weight and blood pressure control without use of antihypertensive drugs. If the captopril test is positive, a renin-angiotensin system blocker should be started. The pre-dialysis blood pressure goal in hemodialysis patients should be < 140/90 mm Hg initially and < 130/80 mm Hg at 3-6 months of dialysis.Turkish Society of Hypertension and Renal DiseasesPublication costs for this article were supported by the Turkish Society of Hypertension and Renal Diseases, a nonprofit national organization in Turkey