$K_{2, t+1}$-free graphs containing an optimal number of $K_{t, t}$'s

This paper provides an explicit construction showing that for prime power $t$ and specific $n$, the maximum number of $K_{t,t}$ copies in a $K_{2,t+1}$-free graph is asymptotically $(1 + o(1))\frac{n^2}{2t(t-1)}$, thereby determining the precise constant factor for the generalized Turán number $ex(n, K_{t, t}, K_{2, t+1})$ in these cases.

The Gist

Imagine you are hosting a massive party with $n$ guests. You want to arrange the seating and interactions in a very specific way to maximize the number of "perfect pairs" you can form, but you have a strict rule to follow: no two people can have too many mutual friends.

This paper is about finding the absolute limit of how many "perfect pairs" you can create under that rule.

Here is the breakdown of the paper's story, translated into everyday language:

The Rules of the Game

1. The Guests ($n$): You have a large crowd of people.
2. The "Perfect Pair" ($K_{t,t}$): In math-speak, this is a group where $t$ people on one side are all friends with $t$ people on the other side. Think of it as a perfect dance floor where two groups of $t$ dancers are all holding hands with everyone in the opposite group.
3. The Forbidden Pattern ($K_{2,t+1}$): This is the rule you must avoid. You cannot have two people who share more than $t$ mutual friends. If two people have $t+1$ friends in common, the party is "ruined" (or in math terms, the graph is invalid).

The Big Question: What is the maximum number of these "perfect dance floors" ($K_{t,t}$) you can build in a party of size $n$ without accidentally creating a forbidden "too many mutual friends" situation?

What We Knew Before

Recently, other mathematicians figured out that the answer grows roughly like the square of the number of guests ($n^2$). They proved it exists, but their method was like a magic trick: they used complex geometry and probability to say, "Hey, a party like this could exist," without actually showing you the guest list or the seating chart.

The Author's New Trick

Vladislav Taranchuk (the author) says, "I can do better. I can build the party explicitly."

He doesn't just say the party exists; he writes down the exact rule for who sits next to whom. However, there's a catch: this specific seating chart only works perfectly when the number of guests ($n$) is a specific type of number (related to prime numbers and powers).

The Construction (The "How-To"):
Imagine the guests are points on a giant, multi-dimensional grid.

• The author uses a special mathematical formula (a polynomial) to decide who is friends with whom.
• The Rule: Two people are friends if their coordinates satisfy a specific equation involving that formula.
• The Result:
  • If you pick any two people, they will share at most $t$ mutual friends. (So, the forbidden pattern never happens).
  • Because of how the formula is built, you end up with a massive number of those "perfect dance floors" ($K_{t,t}$).

The Big Discovery

The author proves that for these specific party sizes, the number of perfect dance floors you can create is almost exactly:
$$\frac{n^2}{2t(t-1)}$$

This is a huge deal because it matches the theoretical maximum limit (the "ceiling") that mathematicians had calculated earlier. It's like saying, "We thought the best we could do was 100 points, and here is a team that actually scores 100 points."

Why This Matters (In Simple Terms)

1. From "Maybe" to "Here": Previous work said, "A perfect party is possible." This paper says, "Here is the exact blueprint for the perfect party."
2. Efficiency: The author shows that you can't do much better than this. The number of perfect pairs you get is as high as mathematically possible, minus a tiny, negligible amount.
3. The Conjecture: The author believes this formula is actually the answer for all party sizes, not just the special ones he built. He thinks the "ceiling" he found is the true limit for everyone.

The Analogy Summary

Think of it like building a bridge.

• Previous work proved that a bridge strong enough to hold a certain weight could exist, using abstract physics.
• This paper builds the bridge using specific steel beams and bolts. It proves that for certain lengths, this bridge holds the maximum weight possible without collapsing.
• The author is confident that this design is the best possible design for bridges of any length, even if he only built the proof for specific lengths so far.

In a nutshell: The author found a precise, step-by-step recipe to create a network of connections that is as crowded with "perfect groups" as possible, without ever breaking the rule about "too many mutual friends."

Technical Summary

Technical Summary: K2,t+1-free Graphs Containing an Optimal Number of Kt,t's

Problem Statement
The paper addresses the generalized Turán number, denoted as $\text{ex}(n, H, F)$, which represents the maximum number of copies of a graph $H$ that can exist in an $F$-free graph on $n$ vertices. Specifically, the authors investigate the case where $H = K_{t,t}$ and $F = K_{2,t+1}$. The goal is to determine the asymptotic behavior of $\text{ex}(n, K_{t,t}, K_{2,t+1})$.

While recent work by Pohoata, Tidor, and Yu established that $\text{ex}(n, K_{t,t}, K_{2,t+1}) = \Theta_t(n^2)$ for all integers $t \ge 3$, their proof relied on probabilistic methods and subspace-evasive sets, yielding a lower bound with a coefficient that was not necessarily optimal. This paper seeks to provide a precise asymptotic formula for specific values of $n$ and $t$ using an explicit construction.

Methodology
The authors employ a deterministic, algebraic construction of graphs based on finite fields and linearized polynomials.

1. Graph Construction ($\Gamma_{t,e}$):

   • Let $t$ be a prime power and $e > 1$ be an integer. Define $q = t^e$.
   • Let $L$ be an $\mathbb{F}_t$-linearized polynomial over $\mathbb{F}_q[x]$ such that the dimension of its kernel is 1. The image space $R_L$ has size $t^{e-1}$.
   • The vertex set is defined as $V = \mathbb{F}_q \times R_L$, resulting in $n = q \cdot t^{e-1} = t^{2e-1}$ vertices.
   • Two distinct vertices $(v_1, v_2)$ and $(w_1, w_2)$ are adjacent if and only if $v_2 + w_2 = L(v_1 w_1)$. Self-loops implied by the equation are removed to ensure a simple graph.

2. Upper Bound Derivation:

   • The authors first establish a precise upper bound for $\text{ex}(n, K_{t,t}, K_{2,t+1})$.
   • The argument relies on the fact that in a $K_{2,t+1}$-free graph, any two vertices share at most $t$ common neighbors.
   • By counting pairs of vertices and analyzing how many times a $K_{t,t}$ is counted via its constituent pairs, they derive the bound:
     $$\text{ex}(n, K_{t,t}, K_{2,t+1}) \le \frac{\binom{n}{2}}{2\binom{t}{2}}$$

3. Structural Analysis of the Construction:

   • $K_{2,t+1}$-freeness: The authors prove that $\Gamma_{t,e}$ contains no $K_{2,t+1}$. For any two vertices, the number of common neighbors is determined by the number of solutions to a linear equation involving $L$. Since $L$ is a $t$-to-1 mapping, there are at most $t$ common neighbors.
   • $K_{t,t}$ Formation: The core of the proof involves analyzing non-adjacent vertices. The authors show that any two non-adjacent vertices with distinct first coordinates uniquely define two sets, $M$ and $N$, of cardinality $t$.
     • $N$ is the set of common neighbors (an $\mathbb{F}_t$-line).
     • $M$ is the $\mathbb{F}_t$-line containing the two vertices.
     • Every vertex in $M$ is algebraically adjacent to every vertex in $N$.
   • If $M \cap N = \emptyset$, the subgraph induced by $M \cup N$ is a $K_{t,t}$. If $|M \cap N| = 1$, the structure forms a tripartite complete graph $K_{1, t-1, t-1}$.

Key Contributions and Results
The primary contribution is the explicit construction of a graph family that achieves the theoretical upper bound for the number of $K_{t,t}$ copies, modulo lower-order terms, for an infinite set of $n$.

• Theorem 1.1: For $t$ being a prime power and $n = t^{2e-1}$ (where $e > 1$ is an integer), the generalized Turán number satisfies:
  $$\text{ex}(n, K_{t,t}, K_{2,t+1}) = (1 + o(1)) \frac{n^2}{2t(t-1)}$$
• Comparison with Previous Bounds: The paper notes that the previous best lower bound from [5] was of the form $(1-o(1)) \frac{n^2}{4t^2 t!}$. The new result improves this coefficient significantly for prime powers $t > 5$.
• Negligibility of Non-Optimal Cases: The authors demonstrate that the number of configurations resulting in $K_{1, t-1, t-1}$ (where $|M \cap N| = 1$) is negligible ($O(n)$) compared to the total count of $K_{t,t}$s ($O(n^2)$), ensuring the asymptotic dominance of the $K_{t,t}$ count.

Significance and Claims
The paper claims to provide a "simple and completely explicit construction" for an infinite, albeit sparse, set of $n$'s, contrasting with the probabilistic existence proofs previously available. The construction is isomorphic to graphs previously studied by Lazebnik and Mubayi in the context of multicolor Ramsey numbers, though this paper applies them specifically to the generalized Turán problem.

The authors conclude by conjecturing that the derived upper bound is tight for all $n$, not just the specific prime power cases constructed:
$$\text{ex}(n, K_{t,t}, K_{2,t+1}) = (1 + o(1)) \frac{n^2}{2t(t-1)}$$
The work serves to refine the understanding of the growth rate of generalized Turán numbers for bipartite graphs, moving from order-of-magnitude estimates to precise asymptotic constants for specific regimes.