[1]
Akama, Y., Berardi, S., Hayashi, S., and Kohlenbach, U.,
An arithmetical hierarchy of the law of excluded middle and related principles
, 19th IEEE Symposium on Logic in Computer Science (LICS 2004), IEEE Computer Society, 2004, pp. 192–201.
[2]
Argyros, S. A. and Todorcevic, S., Ramsey Methods in Analysis, Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, Basel, 2005.
[3]
Berardi, S. and Steila, S.,
Ramsey Theorem for pairs as a classical principle in intuitionistic arithmetic
, 19th International Conference on Types for Proofs and Programs, TYPES 2013 (Matthes, R. and Schubert, A., editors), LIPIcs, vol. 26, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl, 2014, pp. 64–83.
[4]
Berardi, S. and Steila, S.,
An intuitionistic version of Ramsey’s Theorem and its use in Program Termination
. Annals of Pure and Applied Logic, vol. 166 (2015), no. 12, pp. 1382–1406.
[5]
Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, NY, 1967.
[6]
Brattka, V. and Rakotoniaina, T., On the uniform computational content of Ramsey’s Theorem, 2016, arXiv:1508.00471.
[7]
Büchi, J. R.,
On a decision method in restricted second order arithmetic
, Logic, Methodology and Philosophy of Science (Proceeding of the 1960 International Congress) (Nagel, E., Suppes, P., and Tarsk, A., editors), Stanford University Press, Stanford, 1962, pp. 1–11.
[8]
Coquand, T.,
A direct proof of Ramsey’s Theorem
, Author’s website, revised in 2011, 1994.
[9]
Dorais, F. G., Dzhafarov, D. D., Hirst, J. L., Mileti, J. R., and Shafer, P.,
On uniform relationships between combinatorial problems
. Transactions of the American Mathematical Society, vol. 368 (2016), no. 2, pp. 1321–1359.
[10]
Gasarch, W. I.,
Proving programs terminate using well-founded orderings
, Ramsey’s Theorem, and matrices. Advances in Computers, vol. 97 (2015), pp. 147–200.
[11]
Jockusch, C. G.,
Ramsey’s Theorem and recursion theory
, this JOURNAL, vol. 37 (1972), pp. 268–280.
[12]
Kreuzer, A. and Kohlenbach, U.,
Ramsey’s Theorem for pairs and provably recursive functions
. Notre Dame Journal of Formal Logic, vol. 50 (2009), no. 4, pp. 427–444.
[13]
Lee, C. S., Jones, N. D., and Ben-Amram, A. M.,
The size-change principle for program termination
, Conference Record of POPL 2001 (Hankin, C. and Schmidt, D., editors), ACM, New York, NY, 2001, pp. 81–92.
[14]
Podelski, A. and Rybalchenko, A.,
Transition invariants
, 19th IEEE Symposium on Logic in Computer Science (LICS 2004), IEEE Computer Society, 2004, pp. 32–41.
[15]
Ramsey, F. P.,
On a problem in formal logic
. Proceedings of the London Mathematical Society, vol. 30 (1930), pp. 264–286.
[16]
Specker, E.,
Ramsey’s theorem does not hold in recursive set theory
, Logic Colloquium ’69 (Proceedings of the Summer School and Colloquium, Manchester, 1969) (Gandy, R. O. and Yates, C. M. E., editors), North-Holland, Amsterdam, 1971, pp. 439–442.
[17]
Steila, S.,
An intuitionistic analysis of size-change termination
, 20th International Conference on Types for Proofs and Programs, TYPES 2014 (Herbelin, H., Letouzey, P., and Sozeau, M., editors), LIPIcs, vol. 39, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl, 2015, pp. 288–307.
[18]
Vytiniotis, D., Coquand, T., and Wahlstedt, D.,
Stop when you are almost-full - adventures in constructive termination
, Interactive Theorem Proving, ITP 2012 (Beringer, L. and Felty, A. P., editors), Lecture Notes in Computer Science, vol. 7406, Springer, Berlin, Heidelberg, 2012, pp. 250–265.
$k \ge 2$
, Ramsey’s Theorem for pairs and recursive assignments of k colors is equivalent to the Limited Lesser Principle of Omniscience for
${\rm{\Sigma }}_3^0$
formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite k-ary tree there is some
$i < k$
and some branch with infinitely many children of index i.