Erdős-Hajnal Sets and Semigroup Decompositions

1. Erdős-Hajnal Sets and Semigroup Decompositions Joshua N. Cooper Courant Institute

2. Suppose we have a collection of lines in 3D… stack(L)=3 What is the largest stacked subset? Denote the cardinality of the largest stacked subset of a family L of lines stack(L). A set of lines is stacked if it is linearly ordered by the “passing over” relation.

3. There is a natural tournament on L: Define stack(n) = min stack(L). |L| = n Stacked subsets correspond to transitive subtournaments. Ramsey’s Theorem implies stack(n) >> log n.

4. z ℓ2 ℓ1 (a,b,1) z = 1 (a′,b′,1) y z = -1 (c′,d′,-1) (c,d,-1) For a tournament T, define trans(T) to be the size of the largest transitive subtournament. trans(T) is notoriously sensitive to randomness. And line configurations are decidedly not random. ℓ1 passes overℓ2 iff g(a-a′,b-b′,c-c′,d-d′) > 0, where g(x,y,z,w) = (x-z)(xw-yz).

5. R = 2O(n)<<2n(n-1)/2 Re-define a line configuration tournament: V = {v1,…,vn}R4 ∩ E = {(v,w) : g(v-w) > 0} Theorem. [Milnor '64, Thom '65] Let K be the number of ±1-vectors in the set {(r1(x) ,…, rm(x)) : xRm} where rj = sgn ○fj, 1 ≤ i ≤ m, and each fj is a polynomial of degree at most d in k variables. Then R≤(4edm/k)k. d = 3, k = 4n, m = n(n-1)/2 There are very few line configurations compared to the number of tournaments.

6. There is some T which cannot be a subtournament of a line configuration tournament. So what? Definition. A homogeneous subset in a graph is a clique or an independent set. Conjecture [Erdős-Hajnal]. For every graph H, there is an ε> 0 so that every graph on n vertices which has no induced copy of H contains a homogeneous subset of size nε. Alon, Pach, and Solymosi showed that the following is equivalent: Conjecture. For every tournament T, there is an ε> 0 so that every tournament on n vertices which has contains no copy of T as a subtournament contains a transitive subset of size nε. So, if you believe Erdős-Hajnal, every line configuration tournament contains a large transitive subtournament.

7. Define the digraph G( f ), for a k-variable polynomial f, by V = {v1,…,vn}Rk ∩ E = {(v,w) : f(v-w) > 0} Theorem. [Alon, Pach, Pinchasi, Radoičić, Sharir ‘04] Every digraph G( f ) contains a large transitive subtournament, independent set, or complete graph. (Actually, they proved a lot more.) Generalize: Define G(S), for a subset S of Rk, to be the digraph E = {(v,w) : v-wS} V = {v1,…,vn}Rk ∩ These definitions agree when S = f -1(R≥0).

8. Call a subset S of Rk Erdős-Hajnal if G(S) must contain a large transitive subtournament, independent set, or clique. Which other sets are Erdős-Hajnal ? Theorem. Any bounded set S such that 0 ∂Sis Erdős-Hajnal. Either n1/2 points fall in a single square, or lots of points fall into n1/2 squares.

9. A set S Rkis a semigroup if x, ySimplies x+yS. ∩ Theorem. Any semigroup S Rkis Erdős-Hajnal. ∩ b c a b – a S & c – b S c – a S Then G(S) is a quasiorder. Define [a] = {x : a←x←a}, so G(S) induces a partial order on the equivalent classes. Apply Dilworth’s Theorem : Either (A) there is an antichain of size n1/3, or a chain of size n2/3. In the latter case, there is either (B) an equivalence class of size n1/3 or (C) a chain of size n1/3 among elements of equiv. classes. (A) = Independent Set, (B) = Clique, (C) = Transitive Subtournament.

10. Proposition. If S, T are Erdős-Hajnal, then S∩T, S T and Sc are, too. ∩ A semialgebraic set is a subset of Rkdefined by polynomial inequalities. The Alon-Pach-Pinchasi-Radoičić-Sharir Theorem means, in particular, that all semialgebraic sets are Erdős-Hajnal. Might it already be true that all semialgebraic sets belong to the set algebra generated by semigroups? Theorem. When k = 1 or 2, yes. Conjecture. When k = 3, yes. When k > 3, no.

11. The “proof”… The green area is a semigroup.

12. Parting Questions: • Do all 3-dimensional semialgebraic sets belong to the set • algebra generated by semigroups? • What other sets are Erdős-Hajnal? Positive sets of • Chebyshev systems? 3. What is the right exponent for stacked subsets? 1/6 ≤ε≤ 0.565 ≈ log37 4. Is anyone here hiring this year?

13. Thank you!