Rainbow AP(4) in an almost equinumerous coloring

Importance: Medium ✭✭

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Keywords:	arithmetic progression
	rainbow

Recomm. for undergrads: no

Posted	by:	vjungic
	on:	July 20th, 2007

Problem Do 4-colorings of $\mathbb{Z}_{p}$ , for $p$ a large prime, always contain a rainbow $AP(4)$ if each of the color classes is of size of either $\lfloor p/4\rfloor$ or $\lceil p/4\rceil$ ?

It is known that there are equinumerous colorings of $\mathbb{Z}_{4m}$ (i.e. colorings of $\mathbb{Z}_{4m}$ for some $m$ such that each color occurs $m$ times) within which we cannot find rainbow arithmetic progressions of length $4$ . ([CJR])

Bibliography

*[C] David Conlon, Rainbow solutions of linear equations over $\mathbb{Z}_p$ , Discrete Mathematics, 306 (2006) 2056 - 2063.

[CJR] David Conlon, Veselin Jungic, Rados Radoicic, On the existence of rainbow 4-term arithmetic progressions, Graphs and Combinatorics, 23 (2007), 249-254

* indicates original appearance(s) of problem.

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Tight hypergraphs

On September 1st, 2007 Dino says:

It deservs to be mentioned that in any $3$ -colouring of ${\mathbb Z}_p^*/{\mathbb Z}_3^*$ , the equation $x+y=z$ , have an heterochromatic (rainbow) solution; here, ${\mathbb Z}_p^*$ denotes the multiplicative group of the field ${\mathbb Z}_p$ .