On restricted families of projections in R^3

We study projections onto non-degenerate one-dimensional families of lines and planes in $\mathbb{R}^{3}$. Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most $1/2$-dimensional sets $B \subset \mathbb{R}^{3}$ is typically preserved under one-dimensional families of projections onto lines. We improve the result by an $\varepsilon$, proving that if $\dim_{\mathrm{H}} B = s > 1/2$, then the packing dimension of the projections is almost surely at least $\sigma(s) > 1/2$. For projections onto planes, we obtain a similar bound, with the threshold $1/2$ replaced by $1$. In the special case of self-similar sets $K \subset \mathbb{R}^{3}$ without rotations, we obtain a full Marstrand type projection theorem for one-parameter families of projections onto lines. The $\dim_{\mathrm{H}} K \leq 1$ case of the result follows from recent work of M. Hochman, but the $\dim_{\mathrm{H}} K > 1$ part is new: with this assumption, we prove that the projections have positive length almost surely.Comment: 33 pages. v2: small changes, including extended introduction and additional references. To appear in Proc. London Math. So

Colouring set families without monochromatic k-chains

A coloured version of classic extremal problems dates back to Erd\H{o}s and Rothschild, who in 1974 asked which $n$-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erd\H{o}s' extension to $k$-chain-free families. Given a family $\mathcal{F}$ of subsets of $[n]$, we define an $(r,k)$-colouring of $\mathcal{F}$ to be an $r$-colouring of the sets without any monochromatic $k$-chains $F_1 \subset F_2 \subset \dots \subset F_k$. We prove that for $n$ sufficiently large in terms of $k$, the largest $k$-chain-free families also maximise the number of $(2,k)$-colourings. We also show that the middle level, $\binom{[n]}{\lfloor n/2 \rfloor}$, maximises the number of $(3,2)$-colourings, and give asymptotic results on the maximum possible number of $(r,k)$-colourings whenever $r(k-1)$ is divisible by three.Comment: 30 pages, final versio