Unavoidable Multicoloured Families of Configurations

Balogh and Bollob\'as [{\em Combinatorica 25, 2005}] prove that for any $k$ there is a constant $f(k)$ such that any set system with at least $f(k)$ sets reduces to a $k$-star, an $k$-costar or an $k$-chain. They proved $f(k)<(2k)^{2^k}$. Here we improve it to $f(k)<2^{ck^2}$ for some constant $c>0$. This is a special case of the following result on the multi-coloured forbidden configurations at 2 colours. Let $r$ be given. Then there exists a constant $c_r$ so that a matrix with entries drawn from $\{0,1,...,r-1\}$ with at least $2^{c_rk^2}$ different columns will have a $k\times k$ submatrix that can have its rows and columns permuted so that in the resulting matrix will be either $I_k(a,b)$ or $T_k(a,b)$ (for some $a\ne b\in \{0,1,..., r-1\}$), where $I_k(a,b)$ is the $k\times k$ matrix with $a$'s on the diagonal and $b$'s else where, $T_k(a,b)$ the $k\times k$ matrix with $a$'s below the diagonal and $b$'s elsewhere. We also extend to considering the bound on the number of distinct columns, given that the number of rows is $m$, when avoiding a $t k\times k$ matrix obtained by taking any one of the $k \times k$ matrices above and repeating each column $t$ times. We use Ramsey Theory.Comment: 16 pages, add two application

Existence of Good Sweepouts on Closed Manifolds

In this note we establish estimates for the harmonic map heat flow from $S^1$ into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.Comment: 7 pages; added reference; corrected typo

A Covert Data Transport Protocol

Both enterprise and national firewalls filter network connections. For data forensics and botnet removal applications, it is important to establish the information source. In this paper, we describe a data transport layer which allows a client to transfer encrypted data that provides no discernible information regarding the data source. We use a domain generation algorithm (DGA) to encode AES encrypted data into domain names that current tools are unable to reliably differentiate from valid domain names. The domain names are registered using (free) dynamic DNS services. The data transmission format is not vulnerable to Deep Packet Inspection (DPI).Comment: 8 pages, 10 figures, conferenc