Erdös-Heilbronn problem

Let be an Abelian group and let . For , let

(Here, denotes the cardinality of a set .) Let be a prime number and let . It was conjectured by P. Erdös and H. Heilbronn that .

This conjecture, mentioned in [a5], was first proved in [a3], using linear algebra. As a consequence of the lower bound on the degree of the minimal polynomial of the Grasmann derivative, the following theorem is true [a3]: Let be a prime number and let . Then

Applying this theorem with , one obtains the Erdös–Heilbronn conjecture mentioned above. A generalization of the theorem has been obtained in [a2]. Presently (1996), almost nothing is known for composite numbers. The following conjecture is proposed here: Let be a composite number (cf. also Prime number) and let be such that . Then for some .

For a prime number , the above conjecture is an easy consequence of the theorem above. Some applications to integer subset sums are contained in [a6]. Along the same lines, the conjecture has several implications. In particular, for one finds: Let be such that . Then there is a such that and .

This was conjectured partially by Erdös and R. Graham [a5] and follows easily by applying the conjecture twice, after adding .

References

[a1]	N. Alon, "Subset sums" J. Number Th. , 27 (1987) pp. 196–205
[a2]	N. Alon, M.B. Nathanson, I. Z. Rusza, "The polynomial method and restricted sums of congruence classes" J. Number Th. (to appear)
[a3]	J.A. Dias da Silva, Y.O. Hamidoune, "Cyclic subspaces of Grassmann derivations" Bull. London Math. Soc. , 26 (1994) pp. 140–146
[a4]	P. Erdös, H. Heilbronn, "On the addition of residue classes mod " Acta Arith. , 9 (1964) pp. 149–159
[a5]	P. Erdös, R. Graham, "Old and new problems and results in combinatorial number theory" L'Enseign. Math. (1980) pp. 1–128
[a6]	Y.O. Hamidoune, "The representation of some integers as a subset sum" Bull. London Math. Soc. , 26 (1994) pp. 557–563
[a7]	H.B. Mann, "Addition theorems" , R.E. Krieger (1976) (Edition: Second)