## DIMACS TR: 2001-04

## Set-Systems with Restricted Multiple Intersections and Explicit Ramsey Hypergraphs

### Authors: Vince Grolmusz

**
ABSTRACT
**

We give generalizations for the Deza-Frankl-Singhi Theorem in
case of multiple intersections. More exactly, we prove, that if ${\cal
H}$ is a set-system, which satisfies that for some $k$, that the
$k$-wise intersections occupy only $\ell$ residue-classes modulo a $p$
prime, while the sizes of the members of ${\cal H}$ are not in these
residue classes, then the size of ${\cal H}$ is at most
$$(p-1)(k-1)\sum_{i=1}^{|L|}{n\choose i}$$
This result strengthtens a result of F\"uredi (1983), and gives partial
answer to a question of T. S\'os (1976).
As an application, we give explicit constructions for (multi-colored)
Ramsey hypergraphs. By our best knowledge, this is the first explicit
construction of a Ramsey-hypergraph in the literature.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2001/2001-04.ps.gz

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