Stefan David

• MMath
• PhD Student at University of Cambridge

In this paper we introduce and investigate a new type of automata which turns out to be rich in deep and complex phenomena. For our model, a maze is a countable strongly connected digraph called the board together with a proper colouring of its edges (the edges leaving a vertex have distinct colours) and two special vertices: the origin and the des...

For two graphs B and H the strong Ramsey game R(B,H) on the board B and with target H is played as follows. Two players alternately claim edges of B. The first player to build a copy of H wins. If none of the players win, the game is declared a draw. A notorious open question of Beck (1996, 2002, 2011) asks whether the first player has a winning st...

For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on the board $B$ and with target $H$ is played as follows. Two players alternately claim edges of $B$. The first player to build a copy of $H$ wins. If none of the players win, the game is declared a draw. A notorious open question of Beck asks whether the first player has a winni...

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is "taut", i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \ldots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of...

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is "taut", i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \ldots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of...

We propose a geometric framework to study a continuous analogue of symmetric chain decompositions. Our framework is then applied to study symmetric chain decompositions on families of posets arising from lattice points of denominator $n$ inside rational polytopes. Conditions are described which allow us to turn a symmetric chain decomposition of a...