Number Theory » Additive N.T.

The Erdos-Turan conjecture on additive bases ★★★★

Author(s): Erdos; Turan

Let $ B \subseteq {\mathbb N} $. The representation function $ r_B : {\mathbb N} \rightarrow {\mathbb N} $ for $ B $ is given by the rule $ r_B(k) = \#\{ (i,j) \in B \times B : i + j = k \} $. We call $ B $ an additive basis if $ r_B $ is never $ 0 $.

Conjecture   If $ B $ is an additive basis, then $ r_B $ is unbounded.

Keywords: additive basis; representation function

Posted by mdevos
updated June 8th, 2007
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Combinatorics » Matroid Theory

Rota's unimodal conjecture ★★★

Author(s): Rota

Let $ M $ be a matroid of rank $ r $, and for $ 0 \le i \le r $ let $ w_i $ be the number of closed sets of rank $ i $.

Conjecture   $ w_0,w_1,\ldots,w_r $ is unimodal.
Conjecture   $ w_0,w_1,\ldots,w_r $ is log-concave.

Keywords: flat; log-concave; matroid

Posted by mdevos
updated June 8th, 2007
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