We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey's Theorem for colorings of pairs could help in simplifying the definitions of some ∆ 0 2 sets, while some others could not. We also investigate some consequences of Ramsey's Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathematics, including strengthening of known theorems in a more uniform way and also new theorems.2010 Mathematics Subject Classification. 03B30, 03F35.

A graph G is said to be determined by its generalized spectrum (DGS for short) if for any graph H, H and G are cospectral with cospectral complements implies that H is isomorphic to G.It turns out that whether a graph G is DGS is closely related to the arithmetic properties of its walk-matrix. More precisely, let A be the adjacency matrix of a graph G, and let W = [e, Ae, A 2 e, · · · , A n−1 e] (e is the all-one vector) be its walk-matrix. Denote by G n the set of all graphs on n vertices with det(W ) = 0. In [Wang, Generalized spectral characterization of graphs revisited, The Electronic J. Combin., 20 (4),(2013), #P 4 ], the author defined a large family of graphsis square-free and 2 n/2+1 | det(W )} (which may have positive density among all graphs, as suggested by some numerical experiments) and conjectured every graph in F n is DGS. In this paper, we show that the conjecture is actually true, thereby giving a simple arithmetic condition for determining whether a graph is DGS.

In this paper, we prove that for every bumpy Finsler n-sphere (S n , F ) with reversibility λ and flag curvature K satisfying λ λ+1 2 < K ≤ 1, there exist 2[ n+1 2 ] prime closed geodesics. This gives a confirmed answer to a conjecture of D. V. Anosov [Ano] in 1974 for a generic case.

We study the pigeonhole principle for Σ 2 -definable injections with domain twice as large as the codomain, and the weak König lemma for ∆ 0 2 -definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of 2-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for Σ 2 -definable injections.

No abstract