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

A local homology theory for linearly compact modules

We introduce a local homology theory for linearly compact modules which is in some sense dual to the local cohomology theory of A. Grothendieck. Some basic properties such as the noetherianness, the vanishing and non-vanishing of local homology modules of linearly compact modules are proved. A duality theory between local homology and local cohomology modules of linearly compact modules is developed by using Matlis duality and Macdonald duality. As consequences of the duality theorem we obtain some generalizations of well-known results in the theory of local cohomology for semi-discrete linearly compact modules.Comment: 24 page