Balanced supersaturation for some degenerate hypergraphs

A classical theorem of Simonovits from the 1980s asserts that every graph $G$ satisfying ${e(G) \gg v(G)^{1+1/k}}$ must contain $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$ copies of $C_{2k}$. Recently, Morris and Saxton established a balanced version of Simonovits' theorem, showing that such $G$ has $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$ copies of $C_{2k}$, which are `uniformly distributed' over the edges of $G$. Moreover, they used this result to obtain a sharp bound on the number of $C_{2k}$-free graphs via the container method. In this paper, we generalise Morris-Saxton's results for even cycles to $\Theta$-graphs. We also prove analogous results for complete $r$-partite $r$-graphs.Comment: Changed title, abstract and introduction were rewritte

Community-Supported Financing / how transparency on the web promotes good food production

Ethical and educational thoughts on Financing and Investment of small scale/ high quality food producers. Examples of distribution channels in the physical and online world of food commerce

Extremal graph colouring and tiling problems

In this thesis, we study a variety of different extremal graph colouring and tiling problems in finite and infinite graphs. Confirming a conjecture of Gyárfás, we show that for all k, r ∈ N there is a constant C > 0 such that the vertices of every r-edge-coloured complete k-uniform hypergraph can be partitioned into a collection of at most C monochromatic tight cycles. We shall say that the family of tight cycles has finite r-colour tiling number. We further prove that, for all natural numbers k, p and r, the family of p-th powers of k-uniform tight cycles has finite r-colour tiling number. The case where k = 2 settles a problem of Elekes, Soukup, Soukup and Szentmiklóssy. We then show that for all natural numbers ∆, r, every family F = {F1, F2, . . .} of graphs with v (Fn) = n and ∆(Fn) ≤ ∆ for every n ∈ N has finite r-colour tiling number. This makes progress on a conjecture of Grinshpun and Sárközy. We study Ramsey problems for infinite graphs and prove that in every 2-edge- colouring of KN, the countably infinite complete graph, there exists a monochromatic infinite path P such that V (P) has upper density at least (12 + √8)/17 ≈ 0.87226 and further show that this is best possible. This settles a problem of Erdős and Galvin. We study similar problems for many other graphs including trees and graphs of bounded degree or degeneracy and prove analogues of many results concerning graphs with linear Ramsey number in finite Ramsey theory. We also study a different sort of tiling problem which combines classical problems from extremal and probabilistic graph theory, the Corrádi–Hajnal theorem and (a special case of) the Johansson–Kahn–Vu theorem. We prove that there is some constant C > 0 such that the following is true for every n ∈ 3N and every p ≥ Cn−2/3 (log n)1/3. If G is a graph on n vertices with minimum degree at least 2n/3, then Gp (the random subgraph of G obtained by keeping every edge independently with probability p) contains a triangle tiling with high probability

Evaluation of several pre-clinical tools for identifying characteristics associated with limb bone fracture in thoroughbred racehorses

Catastrophic skeletal fractures in racehorses are devastating not only to the animals, owners and trainers, but also to the perception of the sport in the public eye. The majority of these fatal accidents are unlikely to be due to chance, but are rather an end result failure from stress fractures. Stress fractures are overuse injuries resulting from an accumulation of bone tissue damage over time. Because stress fractures are pathological, it is possible that overt fractures can be predicted and prevented. In this study, third metacarpals (MC3) from 33 thoroughbred racehorse comprised of 8 non-fractured controls and 25 horses that experienced fracture of some limb bone were evaluated for correlative factors for fracture using reference point indentation (RPI; Biodent, Osteoprobe), peripheral quantitative computed tomography (pQCT) and Raman spectroscopy. As measured by RPI, fractured racehorses had reduced indentation distance of the RPI probe on the dorsal surface of the MC3, compared to controls. pQCT analysis revealed that horses that fractured long bones had lower cortical bone mineral density in the distal metaphysis than sesamoid fractured or control horses. Also in the distal metaphysis, horses that fractured their MC3s had greater trabecular and total bone mineral content, as well as greater geometric properties compared to other fracture and control groups. Raman spectroscopy showed that the lateral aspect of horses with MC3 fractures had greater mineral:matrix, carbonate:phosphate and decreased bone remodeling ratios compared to the other fractured and control groups. Several parameters between the two RPI devices were also significantly negatively correlated. This study shows that there are likely correlative factors for fracture using these three types of tools, and that future studies could lead to the development of a predictive model for fracture

A note on diameter-Ramsey sets

A finite set A⊂Rd is called diameter-Ramsey if for every r∈N, there exists some n∈N and a finite set B⊂Rn with diam(A)=diam(B) such that whenever B is coloured with r colours, there is a monochromatic set A′⊂B which is congruent to A. We prove that sets of diameter 1 with circumradius larger than 1/2–√ are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than 135∘ are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher and R\"odl. Furthermore, we deduce that there are simplices which are almost regular but not diameter-Ramsey