Large point-line matchings and small Nikodym sets

Abstract: For any integer $d \geq 2$ and prime power $q$, we construct unexpectedly large induced matchings in the point-line incidence graph of $\mathbb{F}<em>{q}<sup>{d}$ by leveraging a new connection with the Furstenberg-Sárközy problem from arithmetic combinatorics. In particular, we significantly improve the previously well-known baselines when $q$ is prime, showing that $\mathbb{F}</em>{q}<sup>{2}$ contains matchings of size $q<sup>{1.233}$ and $\mathbb{F}<em>{q}<sup>{d}$ contains matchings of size $q^{d-o</em>{d}(1)}$. These results and their proofs have several applications. First, we also obtain new constructions for finite field Nikodym sets in dimension $d \geq 2$, improving recent results of Tao by polynomial factors. For example, when $q$ is prime, we show the existence of Nikodym sets in $\mathbb{F}<em>q<sup>d$ of size $q^d - q^{d - o_d(1)}$. Second, we construct a new minimal blocking set in $\mathrm{PG}(2,q)$, solving a longstanding problem in finite geometry. Third, we obtain new constructions for the minimal distance problem (in $\mathbb{R}<sup>{2}$ and also in higher dimensions), improving a recent result of Logunov-Zakharov. We also obtain analogous results for general finite fields with large characteristics. In particular, in one of our constructions we introduce a new special set of points inside the norm hypersurface in $\mathbb{F}</em>{q}<sup>{d}$, which directly generalizes the classical Hermitian unital and which may be of independent interest for applications.

• The paper introduces a new connection between the point-line incidence graph's induced matching parameter $IM(d, q)$ and the square-difference-free sets of integers, leading to significant improvements in previously known 'baseline' lower bounds by polynomial factors.
• The authors construct point-line matchings and Nikodym sets in finite field settings with applications in minimal blocking sets and minimal separation distance configurations.
• For point-line incidence graphs in planar and higher dimensions, the research provides near-optimal exponents and low-dimensional matching constructions that beat previous results by logarithmic factors.

Overview

This paper, by Hunter, Pohoata, Verstraete, and Zhang (2601.19879), studies the maximum size $IM(d,q)$ of an induced matching in the point-line incidence graph of $\mathbb{F}_q^d$: a collection of point-line pairs $(p_i,\ell_i)$ with $p_i \in \ell_j$ if and only if $i=j$. The central contribution is a new connection between this finite-geometric parameter and the Furstenberg–Sárközy problem on square-difference-free sets of integers. Exploiting this connection, the authors obtain polynomial improvements over the previously known "baseline" lower bounds when $q$ is prime, and near-optimal exponents in high dimensions. These matching constructions in turn yield improved upper-bound constructions for finite-field Nikodym sets, a new large minimal blocking set in $\mathrm{PG}(2,q)$ resolving a longstanding problem, and counterexamples for Euclidean minimal-distance (point-line separation) problems relevant to the Heilbronn triangle program.

The paper's main quantitative results are as follows:

Result	Statement
Planar matchings, prime $q$	$IM(2,q) \gg q^{1.2334}$
High dimension, prime $q$	$\mathbb{F}_q^d$0, $\mathbb{F}_q^d$1
Prime powers, $\mathbb{F}_q^d$2	$\mathbb{F}_q^d$3
General prime powers	$\mathbb{F}_q^d$4, $\mathbb{F}_q^d$5
Nikodym sets, $\mathbb{F}_q^d$6	$\mathbb{F}_q^d$7
Nikodym sets, prime $\mathbb{F}_q^d$8, $\mathbb{F}_q^d$9	$(p_i,\ell_i)$0 for an absolute $(p_i,\ell_i)$1
Minimal blocking sets	$(p_i,\ell_i)$2 for prime $(p_i,\ell_i)$3

Each of these improves prior best bounds by polynomial factors; for instance, the planar baseline was $(p_i,\ell_i)$4, and Tao's recent Nikodym construction gave only constant-factor improvements over the probabilistic baseline.

Background: unitals, Paley graphs, and the baseline bound

When $(p_i,\ell_i)$5 is a square, the Hermitian unital — the set $(p_i,\ell_i)$6 of size $(p_i,\ell_i)$7, together with its tangent lines — gives an induced matching of size $(p_i,\ell_i)$8, which is sharp up to constants by Vinh's Szemerédi–Trotter-type incidence theorem ($(p_i,\ell_i)$9). This construction is fundamentally an extension-field phenomenon and does not apply to non-square $p_i \in \ell_j$0. For prime $p_i \in \ell_j$1, the previous best lower bound was $p_i \in \ell_j$2, obtained by lifting either random sets or independent sets in the Paley graph via a construction of Szőnyi: if $p_i \in \ell_j$3 is independent in the Paley graph, then the set $p_i \in \ell_j$4 with lines $p_i \in \ell_j$5 forms an induced matching of size $p_i \in \ell_j$6, since moving along such a line changes $p_i \in \ell_j$7 by a square.

The obstruction to improving this is the classical square-root barrier for Paley clique numbers: the Delsarte–Hoffman eigenvalue method gives $p_i \in \ell_j$8, sharp for square $p_i \in \ell_j$9, with only Hanson–Petridis' constant-factor improvement $i=j$0 known for primes. The authors note that Proposition 2.1 therefore ties any asymptotic improvement of the upper bound $i=j$1 for prime $i=j$2 to breaking this barrier — a point developed further in their concluding conjectures.

The Ruzsa lift

The key new idea replaces Paley independent sets with integer square-difference-free sets. Using Ruzsa's construction (as sharpened by Beigel–Gasarch and Lewko), there exists a square-difference-free subset $i=j$3 with $i=j$4. The authors then show that for prime $i=j$5, any such $i=j$6 lifts to an induced matching of size $i=j$7: one takes $i=j$8 with $i=j$9, $q$0, and lines through $q$1 of direction $q$2. The identity $q$3 shows that moving along a line changes $q$4 by a perfect square; since all coordinates are bounded integers well below $q$5, congruences lift to integer equalities, and square-difference-freeness forces coincidence. This yields $q$6 for all primes $q$7 — a polynomial improvement over the $q$8 baseline, and notably achieved without any progress on Paley cliques themselves.

Higher dimensions via Waring's problem

For dimension $q$9, the authors generalize the lift using $\mathrm{PG}(2,q)$0-th-power-difference-free sets. The baseline here is $\mathrm{PG}(2,q)$1, from lifting independent sets in Cayley graphs generated by nonzero $\mathrm{PG}(2,q)$2-th powers (Proposition 3.1 constructs, for each admissible tuple $\mathrm{PG}(2,q)$3, a direction vector along which the polynomial $\mathrm{PG}(2,q)$4 shifts by exactly $\mathrm{PG}(2,q)$5).

To beat the baseline, the authors work entirely over $\mathrm{PG}(2,q)$6 before reducing modulo a prime $\mathrm{PG}(2,q)$7. They build a polynomial $\mathrm{PG}(2,q)$8 in $\mathrm{PG}(2,q)$9 variables designed so that every lattice point admits a "shift vector" $q$0 with $q$1; the coefficients of $q$2 are constructed recursively by Waring's theorem (every sufficiently large integer is a signed sum of $q$3 $q$4-th powers). Averaging over shifts selects a level set intersecting a large square-difference-free set $q$5 (with $q$6, $q$7), producing an induced matching in dimension $q$8 of size $q$9 with $IM(2,q) \gg q^{1.2334}$0. Since $IM(2,q) \gg q^{1.2334}$1, this gives $IM(2,q) \gg q^{1.2334}$2. An important consequence: while this does not refute the possibility that $IM(2,q) \gg q^{1.2334}$3, it shows no dimension-independent power saving is possible for prime fields.

Prime powers and norm hypersurfaces

For $IM(2,q) \gg q^{1.2334}$4 with $IM(2,q) \gg q^{1.2334}$5, the authors construct a genuinely new geometric object generalizing the Hermitian unital. They consider the norm hypersurface $IM(2,q) \gg q^{1.2334}$6, where $IM(2,q) \gg q^{1.2334}$7 is the relative norm. Using Lagrange interpolation weights $IM(2,q) \gg q^{1.2334}$8 associated to distinct nodes $IM(2,q) \gg q^{1.2334}$9 satisfying $q$0 for $q$1 and $q$2, they prove that for points on $q$3 whose coordinate norms realize the weight vector, the line $q$4 meets $q$5 only at the base point: the weighted norm sum equals $q$6, vanishing back to $q$7 only at $q$8. Injectivity of the weight map (via Bézout's inequality applied to the power-sum system, using a nonsingularity check via the Jacobian) yields $q$9, hence a matching of size $\mathbb{F}_q^d$00, extended to dimension $\mathbb{F}_q^d$01 by Cartesian products. For $\mathbb{F}_q^d$02 this recovers the Hermitian curve and tangents; for $\mathbb{F}_q^d$03 it produces objects the authors suggest may have independent applications, e.g., in Ramsey theory in the spirit of Mattheus–Verstraëte's use of the O'Nan property of unitals.

Two further arguments complete the prime-power picture: a tensorization lemma ($\mathbb{F}_q^d$04, exploiting the $\mathbb{F}_q^d$05-linear coordinate projections of $\mathbb{F}_q^d$06) handles $\mathbb{F}_q^d$07; and a polynomial-ring analogue of the Waring-based lift, working over $\mathbb{F}_q^d$08 with a shifted-power identity $\mathbb{F}_q^d$09 proved via Boole's summation formula, handles $\mathbb{F}_q^d$10. Combining the three regimes gives $\mathbb{F}_q^d$11 with $\mathbb{F}_q^d$12. The regime not covered by improvements remains $\mathbb{F}_q^d$13 a high power of a small fixed prime, where Guo–Kopparty–Sudan-type lower bounds for Nikodym sets were already strong.

Applications to Nikodym sets and blocking sets

The dictionary between matchings and Nikodym sets is elementary but effective: a set is weak Nikodym if and only if its complement supports an induced matching, and the product of a weak Nikodym set with $\mathbb{F}_q^d$14 is Nikodym one dimension up. Consequently $\mathbb{F}_q^d$15 would imply $\mathbb{F}_q^d$16, and conversely small Nikodym sets in dimension $\mathbb{F}_q^d$17 imply $\mathbb{F}_q^d$18.

Applying the matching results immediately improves Tao's recent constructions (Tao, 11 Nov 2025) by polynomial factors: $\mathbb{F}_q^d$19 for odd prime powers, and $\mathbb{F}_q^d$20 for large $\mathbb{F}_q^d$21. For the plane, where the one-dimensional-lift trick fails, the authors develop a projection mechanism: they extract from the high-dimensional lattice construction a set $\mathbb{F}_q^d$22 with escaping directions from every point of the ambient box, then embed into $\mathbb{F}_q^d$23 via a base-$\mathbb{F}_q^d$24 encoding, obtaining $\mathbb{F}_q^d$25 for prime $\mathbb{F}_q^d$26 — the first polynomial improvement over the $\mathbb{F}_q^d$27 baseline in the non-square case.

Via projective duality, a Nikodym set in $\mathbb{F}_q^d$28 yields a minimal cover of $\mathbb{F}_q^d$29 of size at least $\mathbb{F}_q^d$30: for each affine point $\mathbb{F}_q^d$31, the distinguished line $\mathbb{F}_q^d$32 is the unique member of the covering family containing $\mathbb{F}_q^d$33, so any minimal subcover must retain all of them. This settles a longstanding problem of Szőnyi's school: minimal blocking sets in $\mathbb{F}_q^d$34 of size $\mathbb{F}_q^d$35 exist for every prime $\mathbb{F}_q^d$36, exceeding the classical Bruen–Thas $\mathbb{F}_q^d$37-scale examples (which require square $\mathbb{F}_q^d$38) in a different direction — previously no minimal blocking set larger than $\mathbb{F}_q^d$39 was known for prime $\mathbb{F}_q^d$40 beyond ovoid-derived examples of size $\mathbb{F}_q^d$41.

Minimal distance configurations

The final application concerns the Euclidean minimal distance problem introduced by Cohen–Pohoata–Zakharov in their work on Heilbronn's triangle problem: given $\mathbb{F}_q^d$42 point-line pairs in $\mathbb{F}_q^d$43, how small must $\mathbb{F}_q^d$44 be? The statement $\mathbb{F}_q^d$45 asserts a separation bound of order $\mathbb{F}_q^d$46; $\mathbb{F}_q^d$47 holds by CPZ, and Logunov–Zakharov gave a fractal construction showing $\mathbb{F}_q^d$48 fails for some inexplicit $\mathbb{F}_q^d$49.

The authors provide a clean transfer principle: any lattice configuration $\mathbb{F}_q^d$50 with bounded-slope escape directions ($\mathbb{F}_q^d$51, $\mathbb{F}_q^d$52) maps linearly to a Euclidean configuration in $\mathbb{F}_q^d$53 with pairwise distances $\mathbb{F}_q^d$54. Applying this to their constructions yields two concrete consequences:

1. $\mathbb{F}_q^d$55 is false, substantially improving Logunov–Zakharov with an explicit exponent.
2. A conditional equivalence: if $\mathbb{F}_q^d$56 held for any $\mathbb{F}_q^d$57, then every square-difference-free subset of $\mathbb{F}_q^d$58 has size at most $\mathbb{F}_q^d$59 — i.e., a power-saving upper bound for Furstenberg–Sárközy. Conversely, the Ruzsa lift shows that power savings on $\mathbb{F}_q^d$60 would follow from prime-field savings on $\mathbb{F}_q^d$61.
3. In high dimensions, $\mathbb{F}_q^d$62 fails for $\mathbb{F}_q^d$63, so no bound of the form $\mathbb{F}_q^d$64 survives in sufficiently high dimension.

Limitations and open problems

Several limitations are stated explicitly. First, the method cannot improve Tao's Nikodym bound when $\mathbb{F}_q^d$65 is a high power of a small prime; the norm-hypersurface and Waring-lift constructions both degrade in that regime. Second, the planar Nikodym result requires going through higher dimensions and an embedding trick, and the resulting constant $\mathbb{F}_q^d$66 is ineffective in magnitude (it decays like $\mathbb{F}_q^d$67 for the dimension used). Third, essentially nothing is known about upper bounds for $\mathbb{F}_q^d$68 when $\mathbb{F}_q^d$69: even proving $\mathbb{F}_q^d$70 — stated as Conjecture 9.1 — is open, and the authors' own lower bounds show any proof must exploit $\mathbb{F}_q^d$71-dependent savings. Fourth, the proposed prime-field saving $\mathbb{F}_q^d$72 (Conjecture 9.2) remains open; the authors record that it would imply both a polynomial improvement over the Hanson–Petridis bound for Paley clique numbers and the folklore power-saving bound $\mathbb{F}_q^d$73 for Furstenberg–Sárközy, making it a concrete unifying target. Finally, whether the higher-degree norm-hypersurface objects satisfy local forbidden-configuration properties analogous to the O'Nan property of unitals — and whether these could feed into Ramsey constructions — is left open, as is the correct order of $\mathbb{F}_q^d$74 for point-hyperplane incidences in dimensions $\mathbb{F}_q^d$75 (where the paraboloid construction gives $\mathbb{F}_q^d$76 exactly).

Conclusion

This paper establishes induced matchings in finite-field point-line incidence graphs as a flexible organizing object connecting finite geometry, additive combinatorics, and Euclidean incidence theory. Its technical core — lifting square- and $\mathbb{F}_q^d$77-th-power-difference-free integer sets via polynomials with controlled shift behavior, supplemented by a norm-hypersurface construction generalizing Hermitian unitals — delivers polynomial improvements across the prime and most prime-power regimes, with immediate consequences for Nikodym sets, minimal blocking sets, and minimal-distance counterexamples. The open conjectures it formulates, particularly the prime-field saving $\mathbb{F}_q^d$78, now serve as concrete common strengthenings of the Paley clique problem and the Furstenberg–Sárközy problem.