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Random Turán Problems for Graphs with a Vertex Complete to One Part

Sean Longbrake, Sam Spiro

Abstract

Given a graph $F$, the random Turán problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete bipartite graphs and theta graphs. We greatly expand upon these examples by proving tight bounds for a number of bipartite graphs which have a vertex complete to one part. We also prove new general upper bounds for this problem which in many cases do significantly better than the only previous known general upper bound due to Jiang and Longbrake. Our proofs utilize dependent random choice together with the recent technique of balanced vertex supersaturation in conjunction with hypergraph containers.

Random Turán Problems for Graphs with a Vertex Complete to One Part

Abstract

Given a graph , the random Turán problem asks to determine the maximum number of edges in an -free subgraph of . Prior to this work, the only bipartite graphs with known tight bounds included certain classes of complete bipartite graphs and theta graphs. We greatly expand upon these examples by proving tight bounds for a number of bipartite graphs which have a vertex complete to one part. We also prove new general upper bounds for this problem which in many cases do significantly better than the only previous known general upper bound due to Jiang and Longbrake. Our proofs utilize dependent random choice together with the recent technique of balanced vertex supersaturation in conjunction with hypergraph containers.

Paper Structure

This paper contains 8 sections, 18 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. General Upper Bounds
  4. Tight Examples
  5. The Most General Theorems
  6. Preliminaries
  7. Balanced Supersaturation
  8. Vertex Supersaturation

Key Result

Theorem 1.1

If $\mathrm{ex}(n,K_{r,t})=\Theta(n^{2-1/r})$, then a.a.s. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: A depiction of the graph $F_M$ when $M$ is a triangle with edge multiplicities 1, 2, and 3. By \ref{['thm:multigraph']}, this graph is a new tight example for the random Turán Problem.

Theorems & Definitions (39)

  • Theorem 1.1: morris2016number
  • Theorem 1.2: (Informal) jiang2022balanced
  • Definition 1
  • Conjecture 1.3: mckinley2023random
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3
  • ...and 29 more