Tree-width and dimension

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.Comment: Updates on solutions of problems and on bibliograph

On-line partitioning of width w posets into w^O(log log w) chains

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.Comment: 16 pages, 10 figure

Filling cages: reverse mathematics and combinatorial principles

Nella tesi sono analizzati alcuni principi di combinatorica dal punto di vista della reverse mathematics. La reverse mathematics \ue8 un programma di ricerca avviato negli anni settanta e interessato a individuare l'esatta forza, intesa come assiomi riguardanti l'esistenza di insiemi, di teoremi della matematica ordinaria. --- Dopo una concisa introduzione al tema, \ue8 presentato un algoritmo incrementale per reorientare transitivamente grafi orientati infiniti e pseudo-transitivi. L'esistenza di tale algoritmo implica che un teorema di Ghouila-Houri \ue8 dimostrabile in RCA0. --- Grafi e ordini a intervalli sono la comune tematica della seconda parte della tesi. Un primo capitolo \ue8 dedicato all'analisi di diverse caratterizzazioni di grafi numerabili a intervalli e allo studio della relazione tra grafi numerabili a intervalli e ordini numerabili a intervalli. In questo contesto emerge il tema dell'ordinabilit\ue0 unica di grafi a intervalli, a cui \ue8 dedicato il capitolo successivo. L'ultimo capitolo di questa parte riguarda invece enunciati relativi alla dimensione degli ordini numerabili a intervalli. --- La terza parte ruota attorno due enunciati dimostrati da Rival e Sands in un articolo del 1980. Il primo teorema afferma che ogni grafo infinito contiene un sottografo infinito tale che ogni vertice del grafo \ue8 adiacente ad al pi\uf9 uno o a infiniti vertici del sottografo. Si dimostra che questo enunciato \ue8 equivalente ad ACA0, dunque pi\uf9 forte rispetto al teorema di Ramsey per coppie, nonostante la somiglianza dei due principi. Il secondo teorema dimostrato da Rival e Sands asserisce che ogni ordine parziale infinito con larghezza finita contiene una catena infinita tale che ogni punto dell'ordine \ue8 comparabile con nessuno o con infiniti elementi della catena. Quest'ultimo enunciato ristretto a ordini di larghezza k, per ogni k maggiore o uguale a tre, \ue8 dimostrato equivalente ad ADS. Ulteriori enunciati sono studiati nella tesi.In the thesis some combinatorial statements are analysed from the reverse mathematics point of view. Reverse mathematics is a research program, which dates back to the Seventies, interested to find the exact strength, measured in terms of set-existence axioms, of theorems from ordinary non set-theoretic mathematics. --- After a brief introduction to the subject, an on-line (incremental) algorithm to transitivelly reorient infinite pseudo-transitive oriented graphs is defined. This implies that a theorem of Ghouila-Houri is provable in RCA0 and hence is computably true. --- Interval graphs and interval orders are the common theme of the second part of the thesis. A chapter is devoted to analyse the relative strength of different characterisations of countable interval graphs and to study the interplay between countable interval graphs and countable interval orders. In this context arises the theme of unique orderability of interval graphs, which is studied in the following chapter. The last chapter about interval orders inspects the strength of some statements involving the dimension of countable interval orders. --- The third part is devoted to the analysis of two theorems proved by Rival and Sands in 1980. The first principle states that each infinite graph contains an infinite subgraph such that each vertex of the graph is adjacent either to none, or to one or to infinitely many vertices of the subgraph. This statement, restricted to countable graphs, is proved to be equivalent to ACA0 and hence to be stronger than Ramsey's theorem for pairs, despite the similarity of the two principles. The second theorem proved by Rival and Sands states that each infinite partial order with finite width contains an infinite chain such that each point of the poset is comparable either to none or to infinitely many points of the chain. For each k greater or equal to three, the latter principle restricted to countable poset of width k is proved to be equivalent to ADS. Some complementary results are presented in the thesis