A note on propositional proof complexity of some Ramsey-type statements

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F11%3A00369652" target="_blank" >RIV/67985840:_____/11:00369652 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s00153-010-0212-9" target="_blank" >http://dx.doi.org/10.1007/s00153-010-0212-9</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00153-010-0212-9" target="_blank" >10.1007/s00153-010-0212-9</a>

Result language
angličtina
Original language name
A note on propositional proof complexity of some Ramsey-type statements
Original language description
A Ramsey statement denoted n -> (k)(2)(2) says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formulaRAM(n, k) of size O(n(k)) and with terms ofsize ((k)(2)). Let r(k) be the minimal n for which the statement holds. We prove that RAM(r(k), k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlak's work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4(k), k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>Z - Vyzkumny zamer (s odkazem do CEZ)

Publication year
2011
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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