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Kővári-Sós-Turán theorem for hereditary families

Zach Hunter, Aleksa Milojević, Benny Sudakov, István Tomon

TL;DR

The authors study ex^*(n,H,s), the maximum edges in an n-vertex graph with no induced H and no K_{s,s}, within broad hereditary families and under the K_{s,s}-free constraint. They develop a unified framework showing that for bipartite H with bounded degree on one side, trees, cycles, and the cube admit induced Turán-type bounds matching the non-induced regime up to polynomial factors in s. Key methods combine dependent random choice, hypergraph cleaning to produce rich independent sets, and careful embedding arguments to extend induced restrictions from H to the host graphs. The results yield sharp or near-sharp upper bounds and have applications to Erdős–Hajnal-type questions, induced C_4-free subgraphs with large degree, and average-degree bounds in K_{s,s}-free graphs, offering a pathway toward a broader understanding of induced forbidden substructures in Turán-type problems.

Abstract

The celebrated Kővári-Sós-Turán theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose the systematic study of this phenomenon, and state the conjecture that if $H$ is a bipartite graph, then an induced $H$-free and $K_{s,s}$-free graph cannot have much more edges than an $H$-free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by $k$ on one side, obtaining in all the cases similar bounds as in the non-induced setting. Our results also have applications to the Erdős-Hajnal conjecture, the problem of finding induced $C_4$-free subgraphs with large degree and bounding the average degree of $K_{s, s}$-free graphs which do not contain induced subdivisions of a fixed graph.

Kővári-Sós-Turán theorem for hereditary families

TL;DR

The authors study ex^*(n,H,s), the maximum edges in an n-vertex graph with no induced H and no K_{s,s}, within broad hereditary families and under the K_{s,s}-free constraint. They develop a unified framework showing that for bipartite H with bounded degree on one side, trees, cycles, and the cube admit induced Turán-type bounds matching the non-induced regime up to polynomial factors in s. Key methods combine dependent random choice, hypergraph cleaning to produce rich independent sets, and careful embedding arguments to extend induced restrictions from H to the host graphs. The results yield sharp or near-sharp upper bounds and have applications to Erdős–Hajnal-type questions, induced C_4-free subgraphs with large degree, and average-degree bounds in K_{s,s}-free graphs, offering a pathway toward a broader understanding of induced forbidden substructures in Turán-type problems.

Abstract

The celebrated Kővári-Sós-Turán theorem states that any -vertex graph containing no copy of the complete bipartite graph has at most edges. In the past two decades, motivated by the applications in discrete geometry and structural graph theory, a number of results demonstrated that this bound can be greatly improved if the graph satisfies certain structural restrictions. We propose the systematic study of this phenomenon, and state the conjecture that if is a bipartite graph, then an induced -free and -free graph cannot have much more edges than an -free graph. We provide evidence for this conjecture by considering trees, cycles, the cube graph, and bipartite graphs with degrees bounded by on one side, obtaining in all the cases similar bounds as in the non-induced setting. Our results also have applications to the Erdős-Hajnal conjecture, the problem of finding induced -free subgraphs with large degree and bounding the average degree of -free graphs which do not contain induced subdivisions of a fixed graph.
Paper Structure (12 sections, 11 equations, 1 figure)

This paper contains 12 sections, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Bipartite graphs with bounded degree on one side
  5. Embedding the vertices of degree at most $k$
  6. Finding a rich independent set
  7. Finishing the proof
  8. Lower bounds
  9. Trees, induced $C_4$-free subgraphs and the Erdős-Hajnal conjecture
  10. Cycles and the cube
  11. The cube graph
  12. Concluding remarks

Figures (1)

  • Figure 1: Illustration of the proof of Proposition \ref{['prop:cube']}.