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The seven lines of the Fano plane meet pairwise in one point, and Fisher's bound is tight
Example
The seven lines of the Fano plane may be written as the translates modulo of :
Facts & Assumptions
Given: the seven three-element sets above.
A -uniform family on whose distinct members have constant intersection size has at most members (A -uniform family on with all pairwise intersections of size has at most members).
Verification
Each displayed set has size .
Let . Its nonzero differences are , that is, every nonzero residue modulo exactly once. Hence for distinct translates and , an element lies in their intersection exactly when and both lie in , equivalently when is a nonzero difference of two elements of ; that determines uniquely. So any two distinct displayed sets meet in exactly one point.
So the family has subsets on points with constant pairwise intersection size , and [L1] gives . The bound is therefore tight.
Depends on
- A $k$-uniform family on $[n]$ with all pairwise intersections of size $t<k$ has at most $n$ members
- Fisher's inequality, nonuniform form: distinct nonempty $A_1,\dots,A_m\subseteq[n]$ with $\lvert A_i\cap A_j\rvert=t$ for all $i\ne j$ satisfy $m\le n$
- A finite family of subsets of $[n]$ and its incidence matrix over $F$
- The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §4.1 (standard reference, not scraped)