Related papers: Reducibility in the Four-Color Theorem
There are several approaches for using computers in deriving mathematical proofs. For their illustration, we provide an in-depth study of using computer support for proving one complex combinatorial conjecture -- correctness of a strategy…
Can we acquire apriori knowledge of mathematical facts from the outputs of computer programs? People like Burge have argued (correctly in our opinion) that, for example, Appel and Haken acquired apriori knowledge of the Four Color Theorem…
We present an alternate proof of the fact that given any 4-coloring of the plane there exist two points unit distance apart which are identically colored.
Connection of the Four Color Theorem (FCT) with some operations on trees is described. L.H. Kauffman's theorem about FCT and vector cross product is discussed. Operation of transplantation on trees linked with the move of brackets according…
Acceptable but due to extensive usage of a computer rather unpleasant proof of the famous four color map problem of Francis Guthrie were settled eventually by W. Appel and K. Haken in 1976. Using the same method but shortening the proof…
We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003).
We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not…
This chapter provides a hands-on tutorial on the important technique known as self-reducibility. Through a series of "Challenge Problems" that are theorems that the reader will---after being given definitions and tools---try to prove, the…
We give a new proof of the four-color theorem by exhibiting an unavoidable set of 2822 D-reducible configurations. The existence of such a set had been conjectured by several researchers including Stromquist, Appel and Haken, and Robertson,…
We present a new proof of the known fact that the chromatic number of the plane is at least 5. The main difference of this proof is that it can be verified manually without the help of the computer.
In this note we demonstrate that a number of case-heavy combinatorial proofs in the mathematical phylogenetics literature can be proven more compactly using computational support. We use these techniques to also prove several new…
This appendix for our article, "Almost-rainbow edge-colorings of some small subgraphs", contains the full proof of Theorem 4.1.
In this article we consider mathematical fundamentals of one method for proving inequalities by computer, based on the Remez algorithm. Using the well-known results of undecidability of the existence of zeros of real elementary functions,…
For a plane near-triangulation $G$ with the outer face bounded by a cycle $C$, let $n^\star_G$ denote the function that to each $4$-coloring $\psi$ of $C$ assigns the number of ways $\psi$ extends to a $4$-coloring of $G$. The block-count…
We prove a better coloring theorem for aleph_4 and even aleph_3. This has a general topology consequence.
The main result of this paper is a "colored Tverberg theorem for rainbow-unavoidable complexes". This theorem may be considered as a merging of two theorems: "Tverberg theorem for collectively unavoidable complexes" and "balanced colored…
Fr\'ed\'eric Meunier's question about a multicolored Sperner lemma is addressed, leaving the question of connectivity for the color hypergraphs of such a multicolored simplex. Sperner's lemma asserts the existence of a simplex using all the…
The paper compares two generic techniques for deriving lower bounds and impossibility results in distributed computing. First, we prove a speedup theorem (a-la Brandt, 2019), for wait-free colorless algorithms, aiming at capturing the…
We interpret the number of good four-colourings of the faces of a trivalent, spherical polyhedron as the 2-holonomy of the 2-connection of a fibered category, phi, modeled on Rep(sl(2)) and defined over the dual triangulation, T. We also…
Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is…