Home > Journals > TAG > Vol. 3 (2016) > Iss. 1

#### Article Title

#### Abstract

For a fixed *M* x *N* integer lattice *L*(*M*,*N*), we consider the maximum size* *of a subset *A* of *L*(*M*,*N*) which contains no squares of prescribed side lengths *k*(1),...,*k*(*t*). We denote this size by ex(*L*(*M*,*N*), {*k*(1),...,*k*(*t*)}), and when *t* = 1, we abbreviate this parameter to ex(*L*(*M*,*N*), *k*), where *k* = *k*(1).

Our first result gives an exact formula for ex(*L*(*M*,*N*), *k*) for all positive integers *k*, *M*, and *N*, where ex(*L*(*M*,*N*), *k*) = ((3/4) + *o*(1)) *MN* holds for fixed *k* and diverging *M* and *N*. Our second result identifies a subset *A*_{0} of *L*(*M*,*N*) of size* *at least (2/3)*MN* with the property that, for any integer *k* not divisible by three, *A*_{0} contains no squares of side length *k*. Our third result shows that |*A*_{0}| is asymptotically best possible, in that for all positive integers *M* and *N*, we have ex(*L*(*M*,*N*), {1,2}) < (2/3)*MN* + *O*(*M*+*N*). When *M* = *3m*, our estimates on the error above render exact formulas for ex(*L*(3*m*,3), {1,2}) and ex(*L*(3*m*,6), {1,2}).

#### Recommended Citation

Goldwasser, John; Nagle, Brendan; and Saez, Andres
(2016)
"An Extremal Problem for Finite Lattices,"
*Theory and Applications of Graphs*: Vol. 3
:
Iss.
1
, Article 2.

DOI: 10.20429/tag.2016.030102

Available at:
http://digitalcommons.georgiasouthern.edu/tag/vol3/iss1/2

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