Open questions about Ramsey-type statements in reverse mathematics

Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.Comment: 15 page

On the logical strengths of partial solutions to mathematical problems

We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood [‘Reverse mathematics and a Ramsey-type König's lemma’, J. Symb. Log. 77 (2012) 1272–1280], we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to König's lemma, such as restrictions of König's lemma, Boolean satisfiability problems and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak König's lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak König's lemma is robust in the sense of Montalbán [‘Open questions in reverse mathematics’, Bull. Symb. Log. 17 (2011) 431–454]: it is equivalent to several perturbations. We also clarify the relationship between Ramsey-type weak König's lemma and algorithmic randomness by showing that Ramsey-type weak weak König's lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak König's lemma. This answers a question of Flood

Extra heads and invariant allocations

Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y of \Pi such that translating \Pi by -Y yields a configuration whose law is that of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of \Pi, and prove that there is no shift coupling in which \Pi is a function of \Pi^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R^d) to the points of \Pi. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of \Gamma if and only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law of \Gamma is product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].Comment: Published at http://dx.doi.org/10.1214/009117904000000603 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Matching under Preferences

Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory. Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs. Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process. Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully

60 years of cyclic monotonicity: a survey

The primary purpose of this note is to provide an instructional summary of the state of the art regarding cyclic monotonicity and related notions. We will also present how these notions are tied to optimality in the optimal transport (or Monge-Kantorovich) problem

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