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Forbidden Odds, Binaries, and Factorials
In March 1994, I arrived in Florida Atlantic University for the 25th Southeastern International Conference on Combinatorics, Graph Theory, and...
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Imagining the Real or Realizing the Imaginary: Platonism Versus Imaginism
Undoubtedly, a vast majority of mathematicians are Platonists. They believe that mathematical objects exist “out there” independently of the human...
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Coloring in Space
When in 1958, Paul Erdős learned about the chromatic number of the plane problem, he created a number of related problems, some of which we have...
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Ramsey Theory Before Ramsey: Schur’s Coloring Solution of a Colored Problem and Its Generalizations
Nobody remembered – if anyone even noticed – Hilbert’s 1892 lemma by the time the second Ramseyan type result appears in 1916 in number theory as...
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Whose Conjecture Did Van der Waerden Prove? Two Lives Between Two Wars: Issai Schur and Pierre Joseph Henry Baudet
Bartel L. van der Waerden credits “Baudet” [sic] with conjecturing the result about monochromatic arithmetic progressions. Decades later, Van der...
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How Does One Color Infinite Maps? A Bagatelle
How does one measure fun in mathematics? Certainly not by the length of exposition. This is a short chapter, a bagatelle. I hope nonetheless that you...
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From Pigeonhole Principle to Ramsey Principle
The Infinite Pigeonhole Principle states:
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De Grey’s Construction
I am presenting a slightly edited Aubrey de Grey’s crisp and clear description of his construction [G1], translated from the British to the American...
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Polychromatic Number of the Plane and Results Near the Lower Bound
When a great problem withstands all assaults, mathematicians create many related problems. It gives them something to solve, plus sometimes there is...
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In Search of Van der Waerden: Amsterdam, Year 1945
Following the war’s last “three months, distant from all culture and barbarism” in the Austrian Alps, the Van der Waerdens are liberated by the...
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Chromatic Number of the Plane: The Problem
Our good ole Euclidean plane, don’t we know all about it? What else can there be after Pythagoras and Steiner, Euclid, and Hilbert? In this chapter,...
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Geoffrey Exoo and Dan Ismailescu, or 2 Men for 2 Forbidden Distances
We have a team of a Geombinatorics Editor Geoffrey Exoo and Dan Ismailescu. You may recall Chap. 16 of...
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Chromatic Number of the Plane in Special Circumstances
As you know from Chaps. 4 and 6 , 3 years after Dmitry E....
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What Do the Founding Set Theorists Think About the Foundations?
Kurt Gödel and Paul J. Cohen believed that we would eventually identify all the axioms of set theory and when we have done so, we will no longer be...
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Two Celebrated Problems
Histories of two beautiful coloring problems, The Four-Color Problem (4CP) and The Chromatic Number of the Plane Problem (CNP) have been strikingly...
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Edge-Colored Graphs: Ramsey and Folkman Numbers
In this chapter, we will see that no matter how edges of a complete graph Kn are colored in two or, more generally, finitely many colors (each edge...
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Chromatic Number of the Plane: A Historical Essay
It is natural for one to inquire into the authorship of one’s favorite problem. So, in 1991, I turned to countless articles and books. Some of the...
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The Gallai Theorem
The Gallai Theorem is one of my favorite results in all of mathematics. Surprisingly, it is not widely known even among mathematicians. Its creator...
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Polychromatic Number of the Plane and Results Near the Upper Bound
In section 4, we discussed the polychromatic number χp of the plane and looked at the 1970 paper [Rai] by Dmitry E. Raiskii, in which he was the...
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Applications of the Bergelson–Leibman and the Mordell–Faltings Theorems
To achieve a girth 12 unit-distance graph, Paul O’Donnell alters the set D of allowable constant differences. This changes which sets are in S (i.e.,...