Suppose n, k ≥3 are integers with n ≤2k/2. Prove there exists an n ×n matrix A filled with 0’s and 1’s such that no k ×k submatrix is all 0’s or all 1’s. (a k ×k submatrix is a matrix formed by choosing any k rows and any k columns of A). Hint: Use the probabilistic method.

Suppose n, k ≥3 are integers with n ≤2k/2. Prove there exists an n ×n matrix A filled with 0’s and 1’s such that no k ×k submatrix is all 0’s or all 1’s. (a k ×k submatrix is a matrix formed by choosing any k rows and any k columns of A). Hint: Use the probabilistic method.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

H4. Suppose n, k ≥3 are integers with n ≤2k/2. Prove there exists an n ×n matrix A
filled with 0’s and 1’s such that no k ×k submatrix is all 0’s or all 1’s. (a k ×k submatrix is
a matrix formed by choosing any k rows and any k columns of A). Hint: Use the probabilistic
method.

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