Ramsey numbers for partially-ordered sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

Induced minors and well-quasi-ordering

A graph $H$ is an induced minor of a graph $G$ if it can be obtained from an induced subgraph of $G$ by contracting edges. Otherwise, $G$ is said to be $H$-induced minor-free. Robin Thomas showed that $K_4$-induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without $K_4$ and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 -- 247, 1985]. We provide a dichotomy theorem for $H$-induced minor-free graphs and show that the class of $H$-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if $H$ is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved two decomposition theorems which are of independent interest. Similar dichotomy results were previously given for subgraphs by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502, 1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990]

An improvement of the general bound on the largest family of subsets avoiding a subposet

Let $La(n,P)$ be the maximum size of a family of subsets of $[n]= \{1,2, ..., n \}$ not containing $P$ as a (weak) subposet, and let $h(P)$ be the length of a longest chain in $P$. The best known upper bound for $La(n,P)$ in terms of $|P|$ and $h(P)$ is due to Chen and Li, who showed that $La(n,P) \le \frac{1}{m+1} \left(|P| + \frac{1}{2}(m^2 +3m-2)(h(P)-1) -1 \right) {\binom {n} {\lfloor n/2 \rfloor}}$ for any fixed $m \ge 1$. In this paper we show that $La(n,P) \le \frac{1}{2^{k-1}} (|P| + (3k-5)2^{k-2}(h(P)-1) - 1 ) {n \choose {\lfloor n/2\rfloor} }$ for any fixed $k \ge 2$, improving the best known upper bound. By choosing $k$ appropriately, we obtain that $La(n,P) = O\left( h(P) \log_2\left(\frac{|P|}{h(P)}+2\right) \right) {n \choose \lfloor n/2 \rfloor }$ as a corollary, which we show is best possible for general $P$. We also give a different proof of this corollary by using bounds for generalized diamonds. We also show that the Lubell function of a family of subsets of $[n]$ not containing $P$ as an induced subposet is $O(n^c)$ for every $c>\frac{1}{2}$.Comment: Corrected mistakes, improved the writing. Also added a result about the Lubell function with forbidden induced subposets. The final publication will be available at Springer via http://dx.doi.org/10.1007/s11083-016-9390-