Extremal Problems in Number Theory , Combinatorics And Geometry During my long life

During my long life I wrote many papers on these subjects [1]. There are many fascinating and difficult unsolved problems in all three of these topics. I haye to organize the problems in some order. This is not an easy task and anyway not one of my strong points. In number theory I will mainly discuss questions related to van der Waerden's theorem on long arithmetic progressions and problems in additive number theory. In geometry the questions I want to discuss are either metrical problems, e.g. number of distinct distances which must occur between points in a metric space. The metric space usually will be our familiar E2. I will also discuss incidence problems of points in E2. These problems have a purely combinatorial interpretation too, but the results in Ez are completely different than in the finite geometries. In combinatorics I will discuss Sperner, Bamsey and Turân type problems and will try to emphasize their applications to number theory and geometry. Since I must, after all, remain myself, I can not entirely refrain from stating some old and new problems which, in my opinion, perhaps have been undeservedly neglected. I hope the reader will forgive a very old man for some personal and historical reminiscences but to save space I will try to write only facts which I did not mention elsewhere.

[1]  N. Sloane,et al.  The orchard problem , 1974 .

[2]  J. Beck.  On a problem of K.F. Roth concerning irregularities of point distribution , 1983 .

[3]  R. Guy.  Sets of Integers Whose Subsets Have Distinct Sums , 1982 .

[4]  S. Burr.  Generalized ramsey theory for graphs - a survey , 1974 .

[5]  Paul Erdös,et al.  Extremal problems among subsets of a set , 1974, Discret. Math..

[6]  P. Erdos,et al.  ON SOME APPLICATIONS OF GRAPH THEORY TO NUMBER THEORETIC PROBLEMS DEDICATED TO THE MEMORY OF , 1969 .

[7]  SOCIETATIS,et al.  On divisibility properties of sequences of integers bY , 1968 .

[8]  M. Simonovits.  Extremal Graph Problems , Degenerate Extremal Problems , and Supersaturated Graphs , 2010 .

[9]  N. J. A. Sloane,et al.  On Additive Bases and Harmonious Graphs , 1980, SIAM J. Algebraic Discret. Methods.

[10]  V. Sós.  Surveys in Combinatorics: IRREGULARITIES OF PARTITIONS: RAMSEY THEORY, UNIFORM DISTRIBUTION , 1983 .

[11]  Gill Barequet,et al.  Heilbronn's triangle problem , 2007, SCG '07.

[12]  Fan Chung Graham,et al.  A survey of bounds for classical Ramsey numbers , 1983, J. Graph Theory.

[13]  D. J. Kleitman.  On an extremal property of antichains in partial orders , 1974 .

[14]  Jeffrey C. Lagarias,et al.  On the Density of Sequences of Integers the Sum of No Two of which Is a Square. I. Arithmetic Progressions , 1982, J. Comb. Theory A.

[15]  P. Erdös,et al.  On Some Applications of Graph Theory to Geometry , 1967, Canadian Journal of Mathematics.

[16]  Paul Erdös.  COMBINATORIAL PROBLEMS IN GEOMETRY AND NUMBER THEORY , 1979 .

[17]  P. Erdijs.  ON THE APPLICATION OF COMBINATORIAL ANALYSIS TO MUMBER THEORY GEOMETRY AND ANALYSIS , 1970 .

[18]  Peter Ungar,et al.  2N Noncollinear Points Determine at Least 2N Directions , 1982, J. Comb. Theory, Ser. A.

[19]  T. Motzkin.  The lines and planes connecting the points of a finite set , 1951 .

[20]  E. Sperner.  Ein Satz über Untermengen einer endlichen Menge , 1928 .

[21]  Alexandr V. Kostochka.  A class of constructions for turán’s (3, 4)-problem , 1982, Comb..

[22]  P. Pritchard.  Eighteen primes in arithmetic progression , 1983 .

[23]  P. Erdös.  On extremal problems of graphs and generalized graphs , 1964 .

[24]  F. Behrend,et al.  On Sequences of Numbers not Divisible one by another , 1935 .

[25]  János Komlós,et al.  A Dense Infinite Sidon Sequence , 1981, Eur. J. Comb..

[26]  Paul Erdös.  Some Applications of Ramsey's Theorem to Additive Number Theory , 1980, Eur. J. Comb..

[27]  Endre Szemerédi,et al.  Linear problems in combinatorial number theory , 1975 .

[28]  János Komlós,et al.  On Turán’s theorem for sparse graphs , 1981, Comb..

[29]  Paul Erdös,et al.  On the combinatorial problems which I would most like to see solved , 1981, Comb..