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Konrad Zdanowski

Publications and source records attributed to Konrad Zdanowski.

8 recordsLinked to original sources
arXiv2026

The strength of Ramsey's theorem for $\alpha$-large sets

We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles $\mathsf{RT}^{!\alpha}_k$ asserting that every $k$-coloring of the exactly $\alpha$-large subsets of an infinite $X \subseteq \mathbb{N}$ admits an infinite homogeneous set, where $\alpha$-largeness is defined via systems of fundamental sequences in the style of Ketonen and Solovay. For each countable ordinal $\alpha < \Gamma_0$ and each $k \geq 2$, we prove over $\mathsf{RCA}_0$ that the hierarchy of theorems $\mathsf{RT}^{!\a}_k$ corresponds exactly to the hierarchy of systems axiomatized by closure under transfinite Turing jumps, yielding a fine-grained classification between $\mathsf{ACA}_0$ and $\mathsf{ATR}_0$. Our results extend previous work on the case $\alpha=\omega$ and provide a uniform correspondence between countable indecomposable ordinals below $\Gamma_0$ and natural Ramsey-like theorems.

math.LO↗
arXiv2024

Between proof construction and SAT-solving

The classical satisfiability problem (SAT) is used as a natural and general tool to express and solve combinatorial problems that are in NP. We postulate that provability for implicational intuitionistic propositional logic (IIPC) can serve as a similar natural tool to express problems in Pspace. This approach can be particularly convenient for two reasons. One is that provability in full IPC (with all connectives) can be reduced to provability of implicational formulas of order three. Another advantage is a convenient interpretation in terms of simple alternating automata. Additionally, we distinguish some natural subclasses of IIPC corresponding to the complexity classes NP and co-NP. Our experimental results show that a simple decision procedure requires a significant amount of time only in a small fraction of cases.

cs.LO↗
arXiv2024

Reductions of well-ordering principles to combinatorial theorems

A well-ordering principle is a principle of the form: If $X$ is well-ordered then $F(X)$ is well-ordered, where $F$ is some natural operator transforming linear orders into linear orders. Many important subsystems of Second-order Arithmetic of interest in Reverse Mathematics are known to be equivalent to well-ordering principles. We give a unified treatment for proving lower bounds on the logical strength of various Ramsey-theoretic principles relations using characterizations of the corresponding formal systems in terms of well-ordering principles. Our implications (over $RCA_0$) from combinatorial theorems to $ACA_0$ and $ACA_0^+$ also establish uniform computable reductions of the corresponding well-ordering principles to the corresponding Ramsey-type theorems.

math.LO↗
arXiv2017

New bounds on the strength of some restrictions of Hindman's Theorem

We prove upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem. For example, we show that Hindman's Theorem for sums of length at most 2 and 4 colors implies $\mathsf{ACA}_0$. An emerging {\em leitmotiv} is that the known lower bounds for Hindman's Theorem and for its restriction to sums of at most 2 elements are already valid for a number of restricted versions which have simple proofs and better computability- and proof-theoretic upper bounds than the known upper bound for the full version of the theorem. We highlight the role of a sparsity-like condition on the solution set, which we call apartness.

math.LO↗
arXiv2016

On the Mints Hierarchy in First-Order Intuitionistic Logic

We stratify intuitionistic first-order logic over $(\forall,\to)$ into fragments determined by the alternation of positive and negative occurrences of quantifiers (Mints hierarchy). We study the decidability and complexity of these fragments. We prove that even the $Δ_2$ level is undecidable and that $Σ_1$ is Expspace-complete. We also prove that the arity-bounded fragment of $Σ_1$ is complete for co-Nexptime.

cs.LO↗
arXiv2012

The strength of Ramsey Theorem for coloring relatively large sets

We characterize the computational content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called {\em exactly large} sets. An {\it exactly large} set is a set $X\subset\Nat$ such that $\card(X)=\min(X)+1$. The theorem we analyze is as follows. For every infinite subset $M$ of $\Nat$, for every coloring $C$ of the exactly large subsets of $M$ in two colors, there exists and infinite subset $L$ of $M$ such that $C$ is constant on all exactly large subsets of $L$. This theorem is essentially due to Pudlàk and Rödl and independently to Farmaki. We prove that --- over Computable Mathematics --- this theorem is equivalent to closure under the $ω$ Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem. In addition we give a further characterization in terms of truth predicates over Peano Arithmetic. We conjecture that analogous results hold for larger ordinals.

math.LO↗