[Paper Review] Erd\H os-Ko-Rado theorem for $\{0,\pm 1\}$-vectors

Péter Frankl, Andrey Kupavskii|arXiv (Cornell University)|Oct 13, 2015

Limits and Structures in Graph Theory9 references4 citations

TL;DR

This paper extends the classical Erdös-Ko-Rado theorem to {0,±1}-vectors with exactly k +1s and one -1, determining the maximum size of intersecting families under the condition that no two vectors have a scalar product of -2. The key result reveals a phase transition at n = k², where the standard Erdös-Ko-Rado construction becomes suboptimal, and provides exact formulas for m(n,k,1) in both the range 2k ≤ n ≤ k² and n > k² using extremal set theory and shifting techniques.

ABSTRACT

The main object of this paper is to determine the maximum number of $\{0,\pm 1\}$-vectors subject to the following condition. All vectors have length $n$, exactly $k$ of the coordinates are $+1$ and one is $-1$, $n \geq 2k$. Moreover, there are no two vectors whose scalar product equals the possible minimum, $-2$. Thus, this problem may be seen as an extension of the classical Erd\H os-Ko-Rado theorem. Rather surprisingly there is a phase transition in the behaviour of the maximum at $n=k^2$. Nevertheless, our solution is complete. The main tools are from extremal set theory and some of them might be of independent interest.

Motivation & Objective

• To extend the classical ErdÖs-Ko-Rado theorem to families of {0,±1}-vectors with exactly k coordinates +1 and one coordinate -1.
• To determine the maximum size m(n,k,1) of intersecting families where no two vectors have scalar product -2, the theoretical minimum.
• To identify the threshold at n = k² where the standard ErdÖs-Ko-Rado construction ceases to be optimal.
• To provide a complete characterization of m(n,k,1) for all n ≥ 2k using advanced extremal set theory tools.

Proposed method

• Define m(n,k,1) as the maximum size of a family of {0,±1}-vectors with k +1s and one -1, such that no two vectors have scalar product -2.
• Use shifting techniques to transform families into more structured forms while preserving intersecting properties and vector counts.
• Apply combinatorial bounds on shadow sizes and cross-intersecting families to control the growth of families under constraints.
• Establish recursive inequalities via the construction of extended families in higher dimensions, leading to the recurrence m(n+1,k,1) ≥ m(n,k,1) +  (n choose k).
• Prove that for n ≥ k², equality holds: m(n+1,k,1) = m(n,k,1) +  (n choose k), implying a linear growth phase.
• Use case analysis based on the size of the family of vectors with -1 in a fixed coordinate, applying bounds from extremal set theory to derive contradictions for larger-than-optimal families.

Experimental results

Research questions

• RQ1 What is the maximum size of a family of {0,±1}-vectors with exactly k +1s and one -1, such that no two vectors have scalar product -2, for n ≥ 2k?
• RQ2 Does the classical ErdÖs-Ko-Rado construction remain optimal for m(n,k,1) when n > k²?
• RQ3 How does the extremal family size m(n,k,1) behave asymptotically for n > k², and what is the exact formula?
• RQ4 What structural phase transition occurs at n = k² in the behavior of extremal intersecting families of {0,±1}-vectors?
• RQ5 Can the problem be solved completely for l=1 using shifting and shadow techniques in extremal set theory?

Key findings

• For 2k ≤ n ≤ k², the maximum family size is exactly m(n,k,1) = k × (n-1 choose k), achieved by fixing one coordinate to +1.
• For n > k², the ErdÖs-Ko-Rado construction is no longer optimal, and the maximum size grows linearly with n.
• The recurrence m(n+1,k,1) = m(n,k,1) + (n choose k) holds for all n ≥ k², establishing a linear growth phase.
• The exact formula for n > k² is m(n,k,1) = m(k²,k,1) + ∑_{i=k²}^{n-1} (i choose k), derived from the recurrence.
• The phase transition at n = k² is confirmed by the change in extremal structure: from a star-like construction to a linearly growing family.
• The proof relies on advanced extremal set theory, including shifting, shadow bounds, and case analysis on the size of families with -1 in a fixed coordinate.