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Project: SPRING 2024
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Image: ubuntu2204
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Kernel: Python 3 (system-wide)

from IPython.display import Markdown, Image, display

# Biography of Paul Erdős
biography = """
## Biography of Paul Erdős

Paul Erdos was a renowned mathematician known for his significant contributions to the field of discrete mathematics. Born on March 26, 1913, in Budapest, Hungary, Erdos displayed exceptional mathematical talent from a young age.

He made substantial contributions in various areas of discrete mathematics, including graph theory, combinatorics, and number theory. He collaborated with hundreds of mathematicians around the world, earning him the nickname "The Oddball Mathematician." His unique working style involved traveling extensively and staying with fellow mathematicians, often working on problems together for days or even weeks. One of Erdos's most notable contributions was in the field of graph theory. He worked on a wide range of graph-related problems, such as coloring, connectivity, and Hamiltonian cycles. Erdos introduced several important concepts and conjectures, including the Erdos-Szekeres theorem and the Erdos-Renyi random graph model.

In combinatorics, Erdos made significant advances in Ramsey's theory, which examines the existence of order within large, disordered structures. He formulated the famous Erdos-Szekeres theorem, which states that any sequence of a sufficient length contains either a monotonically increasing or decreasing subsequence. Erdos's influence extended beyond his research. He mentored numerous young mathematicians, inspiring them to pursue their passion for mathematics. His collaborative approach and emphasis on problem-solving greatly impacted the field of discrete mathematics and mathematical research as a whole.

Paul Erdos's dedication and prolific output earned him an extraordinary reputation in the mathematical community. He published over 1,500 papers during his career, making him one of the most prolific mathematicians in history. His immense contributions and unique personality continue to inspire and influence mathematicians to this day.
"""

# To display the biography using Markdown and the picture
display(Markdown(biography))
Image("https://mathshistory.st-andrews.ac.uk/Biographies/Erdos/thumbnail.jpg", width="400")

Biography of Paul Erdős

Paul Erdos was a renowned mathematician known for his significant contributions to the field of discrete mathematics. Born on March 26, 1913, in Budapest, Hungary, Erdos displayed exceptional mathematical talent from a young age.

He made substantial contributions in various areas of discrete mathematics, including graph theory, combinatorics, and number theory. He collaborated with hundreds of mathematicians around the world, earning him the nickname "The Oddball Mathematician." His unique working style involved traveling extensively and staying with fellow mathematicians, often working on problems together for days or even weeks. One of Erdos's most notable contributions was in the field of graph theory. He worked on a wide range of graph-related problems, such as coloring, connectivity, and Hamiltonian cycles. Erdos introduced several important concepts and conjectures, including the Erdos-Szekeres theorem and the Erdos-Renyi random graph model.

In combinatorics, Erdos made significant advances in Ramsey's theory, which examines the existence of order within large, disordered structures. He formulated the famous Erdos-Szekeres theorem, which states that any sequence of a sufficient length contains either a monotonically increasing or decreasing subsequence. Erdos's influence extended beyond his research. He mentored numerous young mathematicians, inspiring them to pursue their passion for mathematics. His collaborative approach and emphasis on problem-solving greatly impacted the field of discrete mathematics and mathematical research as a whole.

Paul Erdos's dedication and prolific output earned him an extraordinary reputation in the mathematical community. He published over 1,500 papers during his career, making him one of the most prolific mathematicians in history. His immense contributions and unique personality continue to inspire and influence mathematicians to this day.

Image in a Jupyter notebook

Reflection

Luke: Overall there is a lot of information about Paul Erdos out there. I did come to notice that its all very biographical and doesn't seem to dive supper in depth into his mathematical work. The stuff I did find about how he contributed to the discrete math world was either a very old article that didn't simplify the math well so most of it went over my head or was behind a pay wall. This made it hard to find enough palletable information about his work to fully describe it. Maybe this will contribute to people who find themselves in the same struggle. They will come across a well formatted, easy to read page that is able to give a biography and quick synopsis of Paul Erdos, may he rest in peace.

Paul Erdos' Contribution to Discrete Mathematics

Graph Theory

Chromatic Graph Theory is the number of colors it takes to color each vertex of a graph, such that each adjacent vertex is a different color. The largest number of colors to color a graphs vertices is called their chromatic number. Up until the 1950s it was mainly looked at through the 4-color conjecture, which states that any graph on a plain requires no more than four colors to color each vertex. Paul Erdos applied this concept to set systems stating that no member of a set system should be monochromatic. He also explored the chromatics of infinite graphs. He was able to prove that the chromatic number of an infinite graph was determined by the upper bound of the chromatic number of its subgraphs.

Combinatorics

Combinatorics, in very simple terms, is the study of finite discrete structures and their properties. One of his many contributions is his contribution to the Ramsey Theory. While working with George Szekeres on a problem the pair of them discovered the Ramsey theory. Unfortunately Ramsey had beat them to the punch and had discovered the theory just five years before them. This did not stop Erdos, he worked on the Ramsey finite theorem and helped turn it into the current Ramsey Theory we have today. Erdos was also able to create a proof that was a lot simpler than Ramsey's original proof.

R(k,l)≤(k+l−2k−1)R(k,l) \leq \left(\begin{array}{cc} k+l-2\\ k-1 \end{array}\right) The above proof is Erdos' proof for the Ramsey Theory.
R(k,l)≤(k+l−2k−1)/logc(k+l)R(k,l) \leq \left(\begin{array}{cc} k+l-2\\ k-1 \end{array}\right)/ log^c(k+l) In 1986 Rodl improved the proof and added the constant C.
R(k,l)≤k−1/2+Alogk(k+l−2k−1)R(k,l) \leq k^{-1/2+A \sqrt{log k}}\left(\begin{array}{cc} k+l-2\\ k-1 \end{array}\right) Then Thomason added the constant A when 2≤l≤k2\leq l \leq k

Some other contributions of his:

  1. Classic theorem of number theory

  2. Was a founder of the study of probabilistic number theory

  3. Approximation theory

  4. Prime number theorem

  5. Erdos number 😉