Arie Bialostocki Explained

Arie Bialostocki
Fields:	Discrete Mathematics, Finite Groups
Workplaces:	University of Idaho
Alma Mater:	Tel Aviv University
Doctoral Advisor:	Marcel Herzog
Known For:	Erdős–Ginzburg–Ziv theorem, Zero-sum Ramsey theory

Arie Bialostocki
Nationality:	American, Israeli
Alma Mater:	Tel-Aviv University, Israel
Occupation:	Mathematician
Employer:	University of Idaho
Known For:	Zero-Sum Ramsey theory

Arie Bialostocki is an Israeli American mathematician with expertise and contributions in discrete mathematics and finite groups.

Education and career

Arie received his BSc, MSc, and PhD (1984) degrees from Tel-Aviv University in Israel.^[1] His dissertation was done under the supervision of Marcel Herzog.^[2] After a year of postdoc at University of Calgary, Canada, he took a faculty position at the University of Idaho, became a professor in 1992, and continued to work there until he retired at the end of 2011.^[3] At Idaho, Arie maintained correspondence and collaborations with researchers from around the world who would share similar interests in mathematics. His Erdős number is 1.^[4] He has supervised seven PhD students and numerous undergraduate students who enjoyed his colorful anecdotes and advice. He organized the Research Experience for Undergraduates (REU) programat the University of Idaho from 1999 to 2003 attractingmany promising undergraduates who themselves have gone on to theiroutstanding research careers.

Mathematics research

Arie has published more than 50 publications.^{[5] [6]} Some of Bialostocki's contributions include:

• Bialostocki^[7] redefined^[8] a 

B -injector in a finite group G to be any maximal nilpotent subgroup B  of G  satisfying d _{2(B)=d} _{2(G)} , where d _{2(X)}  is the largest cardinality of a subgroup of G  which is nilpotent of class at most 2 . Using his definition, it was proved by several authors^{[9] [10] [11] [12]} that in many non-solvable groups the nilpotent injectors form a unique conjugacy class.

• Bialostocki contributed to the generalization of the Erdős-Ginzburg-Ziv theorem (also known as the EGZ theorem).^{[13] [14]} He conjectured: if 

A = (a _{1,a} _{2,\ldots,a} _{n)}  is a sequence of elements of { Z} _{m} , then A  contains at least {\lfloor{n/ 2 }\rfloor \choose {m}} + {\lceil{n/ 2 }\rceil \choose{m}}  zero sums of length m . The EGZ theorem is a special case where n =2 m -1 . The conjecture was partially confirmed by Kisin,^[15] Füredi and Kleitman,^[16] and Grynkiewicz.^[17]

• Bialostocki introduced the EGZ polynomials and contributed to generalize the EGZ theorem for higher degree polynomials.^{[18] [19]} The EGZ theorem is associated with the first degree elementary polynomial.
• Bialostocki and Dierker^{[20] [21]} introduced^[22] the relationship of EGZ theorem to Ramsey Theory on graphs.
• Bialostocki, Erdős, and Lefmann introduced^[23] the relationship of EGZ theorem to Ramsey Theory on the positive integers.
• In Jakobs and Jungnickel's book "Einführung in die Kombinatorik",^[24] Bialostocki and Dierker are attributed for introducing Zero-sum Ramsey theory. In Landman and Robertson's book "Ramsey Theory on the Integers",^[25] the number

b(m, k; r) is defined in honor of Bialostocki's contributions to the Zero-sum Ramsey theory.

• Bialostocki, Dierker, and Voxman^[26] suggested^[27] a conjecture offering a modular strengthening of the Erdős–Szekeres theorem proving that the number of points in the interior of the polygon is divisible by

k , provided that total number of points n\geqslant k +2 . Károlyi, Pach and Tóth^[28] made further progress toward the proof of the conjecture.

• In Recreational Mathematics, Arie's paper on application of elementary group theory to Peg Solitaire is a suggested reading in Joseph Gallian's book^[29] on Abstract Algebra.

Notes and References

1. Bialostocki. Arie. An Application of Elementary Group Theory to Central Solitaire. The College Mathematics Journal. 29. 3. 208–212. 1998. 10.1080/07468342.1998.11973941.
2. at the Mathematics Genealogy Project
3. Web site: Professor Arie Bialostocki retires. 2023-05-22.
4. Bialostocki. Arie. Erdős. Paul. Lefmann. Hanno. Monochromatic and zero-sum sets of nondecreasing diameter . Discrete Mathematics. 137. 1–3. 19–34. 1995. 10.1016/0012-365X(93)E0148-W . free.
5. at zbMATH Open
6. at Google scholar
7. Bialostocki. Arie. Nilpotent injectors in symmetric groups. Israel Journal of Mathematics. 41. 3. 261–273. 1982. 10.1007/BF02771725. 122321992.
8. by A. R. Camina at zbMATH Open
9. Sheu. Tsung-Luen. Nilpotent injectors in general linear groups. Journal of Algebra. 160. 2. 380–418. 1993. 10.1006/jabr.1993.1192 . free.
10. Mohammed. Mashhour Ibrahim. On nilpotent injectors of Fischer group M( 22 ) . Hokkaido Mathematical Journal. 38. 4. 627–633. 2009. 10.14492/hokmj/1258554237 . free.
11. Flavell. Paul. Nilpotent injectors in finite groups all of whose local subgroups are N-constrained. Journal of Algebra. 149. 2. 405–418. 1992. 10.1016/0021-8693(92)90024-G . free.
12. Alali. M. I. M.. Hering. Ch.. Neumann. A.. More on B-injectors of sporadic groups. Communications in Algebra. 28. 4. 2185–2190. 2000. 10.1080/00927870008826951. 120962734.
13. Book: Bialostocki. A.. Lotspeich. M.. Some developments of the Erdős-Ginzburg-Ziv theorem I. Sets, graphs, and numbers: a birthday salute to Vera T. Sós and András Hajnal. Colloquia mathematica Societatis János Bolyai. 97–117. 1992.
14. Bialostocki. Arie. Dierker. Paul. Grynkiewicz. David. Lotspeich. Mark. On some developments of the Erdős-Ginzburg-Ziv theorem II. Acta Arithmetica. 110. 2. 173–184. 2003. 10.4064/aa110-2-7 . 2003AcAri.110..173B. free.
15. Kisin. M.. The number of zero sums modulo m in a sequence of length n. Mathematika. 41. 1. 149–163. 1994. 10.1112/S0025579300007257.
16. Book: Füredi . Z. . Zoltán Füredi. Kleitman . D. J. . Daniel Kleitman. Combinatorics, Paul Erdős is eighty (volume 1). The minimal number of zero sums. Bolyai Society Mathematical Studies . János Bolyai Mathematical Society. 159–172. 1993.
17. Grynkiewicz. David J.. On the number of m -term zero-sum subsequences. Acta Arithmetica. 121. 3. 275–298. 2006. 10.4064/aa121-3-5 . 2006AcAri.121..275G. free.
18. Bialostocki. Arie. Luong. Tran Dinh. Cubic symmetric polynomials yielding variations of the Erdős-Ginzburg-Ziv theorem. Acta Mathematica Hungarica. 142. 152–166. 2014. 10.1007/s10474-013-0346-4. 254240326.
19. Ahmed. Tanbir. Bialostocki. Arie. Pham. Thang. Vinh. Le Anh. Power sum polynomials as relaxed EGZ polynomials. Integers. 19. A49. 2019.
20. Bialostocki. A.. Dierker. P.. Zero sum Ramsey theorems. Congressus Numerantium. 70. 119–130. 1990.
21. Bialostocki. A.. Dierker. P.. On the Erdős-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings. Discrete Mathematics. 110. 1–3. 1–8. 1992. 10.1016/0012-365X(92)90695-C . free.
22. by R. L. Graham at MathSciNet
23. by Ralph Faudree at zbMATH Open
24. Book: Jakobs. Conrad. Jungnickel. Dieter. Einführung in die Kombinatorik . de Gruyter Lehrbuch. 3-11-016727-1. 2004. 10.1515/9783110197990.
25. Book: Landman. Bruce. Robertson. Aaron. Ramsey Theory on the Integers. Second . Student Mathematical Library. 73. American Mathematical Society. 978-0-8218-9867-3. 2015.
26. Bialostocki. Arie. Dierker. P.. Voxman. B.. Some notes on the Erdős-Szekeres theorem . Discrete Mathematics. 91. 3. 231–238. 1991. 10.1016/0012-365X(90)90232-7 . free.
27. by Yair Caro at MathSciNet
28. Károlyi. Gy.. J.. Pach. Tóth. G.. . Studia Scientiarum Mathematicarum Hungarica. 38. 245–259. 2001. 1–4. 10.1556/sscmath.38.2001.1-4.17.
29. Book: Gallian. Joseph A.. Contemporary Abstract Algebra . Cengage Learning. 978-1-305-65796-0. 2015.