Cone dependence -- a basic combinatorial concept (Q1404310)

A set \(A \subset \mathbb{R}^n\) is cone independent of \(B \subset \mathbb{R}^n\) if no \(a \in A\) is a non-negative linear combination of \(B \setminus a.\) If \(B=A\) then \(A\) is called a cone independent set. Denote \(P(n)\) the set of all cone independent \((0,1)\)-vector subsets of \(\mathbb{R}^n.\) Furthermore denote \(P_n(k,\ell)\) the set of all \((0,1)\)-vector subsets of \( \mathbb{R}^n,\) such that every vector in \(A\in P_n(k,\ell)\) has exactly \(k\) ones, and every \((0,1)\)-vector with exactly \(\ell\) ones is cone independent of \(A.\) The aim of this paper is to determine the maximum possible cardinalities \(c(n)\) in \(P(n)\) and \(c_n(k,\ell)\) in \(P_n(k,\ell).\) These new concepts are hard: for example the value \(c_n(k,k+1)\) coincides with the Turán number \(\tau_n(k,k+1).\) The paper proves that \(\lim_n c(n)/2^n\) exists and is in between 0.55 and 1. The conjecture is that the limes is less than 1. The authors determine two special cases of \(c_n(k,\ell)\) and formulate a conjecture for the general \(c_n(k,\ell).\) The proofs use classical results of Turán theory and the shifting operations of extremal set theory.