Tur\'an numbers for $K_{s,t}$-free graphs: topological obstructions and algebraic constructions

TL;DR

This paper explores the limitations of existing constructions for extremal bipartite graphs avoiding complete bipartite subgraphs, revealing topological obstructions and providing new algebraic constructions for larger parameters.

Contribution

It demonstrates topological obstructions to generalizing known extremal graph constructions and introduces new algebraic methods for constructing large $K_{s,t}$-free graphs.

Findings

Hypersurfaces in $\\R^s \\times \\R^s$ contain large grids.

Known constructions for $K_{2,2}$ and $K_{3,3}$-free graphs cannot be extended to $K_{s,s}$ for $s \\geq 4$.

New algebraic constructions for extremal $K_{s,t}$-free graphs for large $t$.

Abstract

We show that every hypersurface in $\R^s\times \R^s$ contains a large grid, i.e., the set of the form $S\times T$, with $S,T\subset \R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free graphs cannot be generalized to a similar construction of $K_{s,s}$-free graphs for any $s\geq 4$. We also give new constructions of extremal $K_{s,t}$-free graphs for large $t$.