Dense triangle-free $(n, d, λ)$-graphs for all orders

TL;DR

This work addresses constructing triangle-free $(n,d,\lambda)$-graphs that are simultaneously highly regular and pseudorandom for all sufficiently large $n$. It blends Alon’s triangle-free construction with a degree-adjusting gadget (via two edge-disjoint subgraphs $R$ and $S$) to produce a $d'$-regular subgraph while preserving a controlled eigenvalue bound through careful deletions and parity adjustments. The main result shows the existence of triangle-free $(n,d',\lambda')$-graphs with $d' \ge \frac{1}{20}n^{2/3}$ and $\lambda' \le 41 n^{1/3}\log^{1/2} n$, implying $\lambda' = O((d'\log n)^{1/2})$ for large $n$. This extends Alon’s density boundary to all large orders up to a polylogarithmic factor and provides a constructive framework that combines pseudorandomness, degree regularization, and parity corrections with potential applications in extremal graph theory and coding theory.

Abstract

In 1994, Alon construct a triangle-free $(n,d,λ)$-graph with $d = Ω(n^{2/3})$ and $λ= O(d^{1/2})$ for an exponentially increasing sequence of integers $n$. Using his ingenious construction, we deduce that there exist triangle-free $(n,d,λ)$-graphs with $d = Ω(n^{2/3})$ and $λ= O( (d \log n)^{1/2} )$ for all sufficiently large $n$.

Theorems & Definitions (18)

• Definition 1.1
• Lemma 1.2: Expander Mixing Lemma
• Theorem 1.3: Alon alon1994explicit
• Theorem 1.4: Conlon conlon2017sequence
• Theorem 1.6
• Lemma 2.1: Chernoff bound janson2011random
• Lemma 2.2
• proof
• Lemma 2.3
• proof