Given a distribution of pebbles on the vertices of a graph G, a pebbling move takes two pebbles from one vertex and puts one on a neighboring vertex. The pebbling number Π(G) is the least k such that for every distribution of k pebbles and every vertex r, a pebble can be moved to r. The optimal pebbling number Π OP T (G) is the least k such that some distribution of k pebbles permits reaching each vertex.Using new tools (such as the "Squishing" and "Smoothing" Lemmas), we give short proofs of prior results on these parameters for paths, cycles, trees, and hypercubes, a new linear-time algorithm for computing Π(G) on trees, and new results on Π OP T

We study an ordered version of hypergraph Ramsey numbers for linearly ordered vertex sets, due to Fox, Pach, Sudakov, and Suk. In the k-uniform ordered path, the edges are the sets of k consecutive vertices in a linear order. Moshkovitz and Shapira described its ordered Ramsey number in terms of an enumerative problem: it equals 1 plus the number of elements in the poset obtained by starting with a certain disjoint union of chains and repeatedly taking the poset of down-sets, k − 1 times. After presenting a proof of this and the resulting bounds, we apply the bounds to study the minimum number of interval graphs whose union is the line graph of the n-vertex complete graph, proving the conjecture of Heldt, Knauer, and Ueckerdt that this grows with n. In fact, the growth rate is between Ω( log log n log log log n ) and O(log log n).MSC codes: 05C55, 05C62, 05C65, 05C35.

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Abstract. Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension of a finite poset is bounded in terms of its height and the tree-width of its cover graph.

When graph Ramsey theory is viewed as a game, "Painter" 2-colors the edges of a graph presented by "Builder". Builder wins if every coloring has a monochromatic copy of a fixed graph $G$. In the on-line version, iteratively, Builder presents one edge and Painter must color it. Builder must keep the presented graph in a class ${\cal H}$. Builder wins the game $(G,{\cal H})$ if a monochromatic copy of $G$ can be forced. The on-line degree Ramsey number $\mathring {R}_\Delta(G)$ is the least $k$ such that Builder wins $(G,{\cal H})$ when ${\mathcal H}$ is the class of graphs with maximum degree at most $k$. Our results include: 1) $\mathring {R}_\Delta(G)\!\le\!3$ if and only if $G$ is a linear forest or each component lies inside $K_{1,3}$. 2) $\mathring {R}_\Delta(G)\ge \Delta(G)+t-1$, where $t=\max_{uv\in E(G)}\min\{d(u),d(v)\}$. 3) $\mathring {R}_\Delta(G)\le d_1+d_2-1$ for a tree $G$, where $d_1$ and $d_2$ are two largest vertex degrees. 4) $4\le \mathring {R}_\Delta(C_n)\le 5$, with $\mathring {R}_\Delta(C_n)=4$ except for finitely many odd values of $n$. 5) $\mathring {R}_\Delta(G)\le6$ when $\Delta(G)\le 2$. The lower bounds come from strategies for Painter that color edges red whenever the red graph remains in a specified class. The upper bounds use a result showing that Builder may assume that Painter plays "consistently".