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Two new results on maximal left-compressed intersecting families

Allan Flower, Richard Mycroft

TL;DR

The paper advances the theory of maximal left-compressed intersecting families (MLCIFs) by establishing that the number of $k$-uniform MLCIFs on a ground set of size $n$ grows doubly exponentially in $k$, with bounds that are tight up to a $\log$-factor in the exponent. It also shows that the $k$ canonical MLCIFs $\langle i \rangle$ are exactly the weight-maximizers under any non-trivial increasing weight function, and that each canonical MLCIF is uniquely optimal for some such $\omega$, thereby generalizing Erdős–Ko–Rado to weighted left-compressed settings. The authors develop boundary-set and type machinery, leverage a one-to-one correspondence with MLCIFs on $[2k]$, and connect to the poset $L(k,k)$ to bound antichains, yielding precise asymptotics for the number of MLCIFs. They also discuss potential improvements via hypergraph containers and a supersaturation conjecture, outlining a path to sharper enumeration results and deeper structural understanding of these families.

Abstract

This paper presents two new results on the theory of maximal left-compressed intersecting families (MLCIFs). First, we answer a question raised by Barber by showing that the number of $k$-uniform MLCIFs on a ground set of size $n$ grows as a doubly-exponential function of $k$, which we identify up to a log factor in the exponent. Among these MLCIFs we identify $k$ specific MLCIFs -- which we call the canonical MLCIFs -- as being in a meaningful way the most important MLCIFs. Specifically, our second main result shows that the canonical MLCIFs are precisely those which can have maximum weight among all $k$-uniform MLCIFs under a non-trivial increasing weight function, and moreover that each canonical MLCIF is the unique $k$-uniform MLCIF of maximum weight for some increasing weight function. This gives an interesting generalisation of the Erdős--Ko--Rado theorem to a notion of size which places greater significance on some elements of the ground set than others.

Two new results on maximal left-compressed intersecting families

TL;DR

The paper advances the theory of maximal left-compressed intersecting families (MLCIFs) by establishing that the number of -uniform MLCIFs on a ground set of size grows doubly exponentially in , with bounds that are tight up to a -factor in the exponent. It also shows that the canonical MLCIFs are exactly the weight-maximizers under any non-trivial increasing weight function, and that each canonical MLCIF is uniquely optimal for some such , thereby generalizing Erdős–Ko–Rado to weighted left-compressed settings. The authors develop boundary-set and type machinery, leverage a one-to-one correspondence with MLCIFs on , and connect to the poset to bound antichains, yielding precise asymptotics for the number of MLCIFs. They also discuss potential improvements via hypergraph containers and a supersaturation conjecture, outlining a path to sharper enumeration results and deeper structural understanding of these families.

Abstract

This paper presents two new results on the theory of maximal left-compressed intersecting families (MLCIFs). First, we answer a question raised by Barber by showing that the number of -uniform MLCIFs on a ground set of size grows as a doubly-exponential function of , which we identify up to a log factor in the exponent. Among these MLCIFs we identify specific MLCIFs -- which we call the canonical MLCIFs -- as being in a meaningful way the most important MLCIFs. Specifically, our second main result shows that the canonical MLCIFs are precisely those which can have maximum weight among all -uniform MLCIFs under a non-trivial increasing weight function, and moreover that each canonical MLCIF is the unique -uniform MLCIF of maximum weight for some increasing weight function. This gives an interesting generalisation of the Erdős--Ko--Rado theorem to a notion of size which places greater significance on some elements of the ground set than others.

Paper Structure

This paper contains 9 sections, 18 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The number of MLCIFs on n elements.
  3. Optimal MLCIFs under increasing weight functions.
  4. Notation
  5. Key ideas
  6. Boundary sets
  7. The type of an MLCIF
  8. MLCIFs under increasing weight functions
  9. The number of MLCIFs

Key Result

Theorem 1

For each integer $k \geqslant 2$ we have $2^{\frac{1}{2}\binom{k-1}{\lfloor k/2\rfloor}} \leqslant |{\mathcal{M}}_{k}| \leqslant 2^{\frac{1}{2}\binom{2k}{k}}$. Moreover, for $k$ sufficiently large we have $2^{\frac{1}{9}k^{-3/2}\binom{2k}{k}} < |{\mathcal{M}}_{k}| < 2^{7\log_2(k)k^{-3/2}\binom{2k}{k

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Lemma 3: barber-max_hitting
  • Lemma 4: barber-max_hitting
  • Proposition 5
  • proof
  • Corollary 6
  • proof
  • Lemma 7
  • proof
  • ...and 21 more