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We present a closed-form solution for n-th term of a general three-term recurrence relation with arbitrary given n-dependent coefficients. The derivation and corresponding proof are based on two approaches, which we develop and describe in…
Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has…
Given a 3-colorable graph $X$, the 3-coloring complex $B(X)$ is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of $X$. Two color classes $C,D \in V(B(X))$ are joined by an edge if $C$…
Chromatic polynomials have been studied extensively, giving us results such as the Fundamental Reduction Theorem and closed formulas for the chromatic polynomials of common classes of graphs. Though, none of those extend to the context of…
We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an…
The computation of matrix functions is a well-studied problem. Of special importance are the exponential and the logarithm of a matrix, where the latter also raises existence and uniqueness questions. This is particularly relevant in the…
We show that there is a $\beta$-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a $\Pi^1_2$-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that…
Ramsey's theorem asserts that every $k$-coloring of $[\omega]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable $k$-coloring of $[\omega]^n$ whose solutions compute the halting set. On the other hand,…
Hadwiger's conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colorable. It is known that the exact version of Hadwiger's conjecture does not extend to list coloring, but it has been conjectured by Kawarabayashi and Mohar…
A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive…
We address the question of the "partition regularity" of the Pythagorean equation a^2+b^2=c^2; in particular, can the natural numbers be assigned a 2-coloring, so that no Pythagorean triple (i.e., a solution to the equation) is…
In previous papers, for an arithmetical function $f_0$, we defined functions $f_m$ and $c_m$ and designated numbers of restricted words over a finite alphabet counted by these functions. In this paper, we examine the reverse problem for…
We study the strength of $\RRT^3_2$, Rainbow Ramsey Theorem for colorings of triples, and prove that $\RCA + \RRT^3_2$ implies neither $\WKL$ nor $\RRT^4_2$. To this end, we establish some recursion theoretic properties of cohesive sets and…
We propose a new proof technique that aims to be applied to the same problems as the Lov\'asz Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve…
In this paper, we investigate the existence of $C^n$, $n\in \mathbb{N}^+$, solutions for a class of second-order iterative functional equations involving iterates of the unknown function and a nonlinear term. Applying the Fiber Contraction…
Hindman proved that, whenever the set $\mathbb{N}$ of naturals is finitely colored, there must exist non-constant monochromatic solution of the equation $a+b=cd$. In this paper we extend this result for dense subsemigroups of $((0, \infty),…
For any complex number $c$, let $\sigma_c\colon\mathbb N\rightarrow\mathbb C$ denote the divisor function defined by $\sigma_c(n)=\displaystyle{\sum_{d|n}d^c}$ for all $n\in\mathbb N$, and define $R(c)=\{\sigma_c(n)\in\mathbb C\colon…
We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs,…
We prove the following result: there is a family $R = \langle R_0,R_1,\ldots \rangle$ of subsets of $\omega$ such that for every stable coloring $c : [\omega]^2 \to k$ hyperarithmetical in $R$ and every finite collection of Turing…
We show that a question of Miller and Solomon -- that whether there exists a coloring $c:d^{<\omega}\rightarrow k$ that does not admit a $c$-computable variable word infinite solution, is equivalent to a natural, nontrivial combinatorial…