Lower Bounds for Small Ramsey Numbers on Hypergraphs

Lower Bounds for Small Ramsey Numbers on Hypergraphs Sixue Liu Princeton University

3/3/2021 2 of 45 Contributions • New lower bounds on hypergraphs: some of them have only trivial lower bounds before. • Several new recurrence relations: derive new lower bounds based on smaller values. • Draw novel connections to satisfiability theory.

3/3/2021 3 of 45 Ramsey Number Definition

3/3/2021 4 of 45 Ramsey Number Known Results A graph on 6 vertices

3/3/2021 5 of 45 Ramsey Number Known Results A graph on 6 vertices No matter how to color, there is either a blue or red triangle

3/3/2021 6 of 45 Ramsey Number Known Results A graph on 6 vertices No matter how to color, there is either a blue or red triangle

3/3/2021 7 of 45 Hypergraph Definition 4 -hypergraph on 5 vertices with 2 hyperedges

3/3/2021 8 of 45 Hypergraph Definition

3/3/2021 9 of 45 Hypergraph Known Results

3/3/2021 10 of 45 Ramsey Number Lower Bounds

3/3/2021 11 of 45 Constructive Lower Bound 1 2 5 4 An uncolored graph on 5 vertices 3 CNF Encoding

3/3/2021 12 of 45 Constructive Lower Bound 1 2 5 4 An uncolored graph on 5 vertices 3 CNF Encoding

3/3/2021 13 of 45 Constructive Lower Bound 1 2 5 4 An uncolored graph on 5 vertices 3 CNF Encoding

3/3/2021 14 of 45 Constructive Lower Bound CNF Encoding

3/3/2021 15 of 45 Constructive Lower Bound SAT Solving 1. Standard SAT solver: Mini. SAT, Glucose • Failed in extremely small graph: 10 vertices 2. Incomplete Solver: UBCSAT, CCASat • Failed in small graph: 25 vertices Need more insights!

3/3/2021 16 of 45 Constructive Lower Bound SAT Solving New Algorithm 1. Generate a random assignment 2. In each iteration: • If no unsat clause, return solution • Else if exists variable whose flipping decreases the number of unsat clause (score>0) • If it’s not forbidden, choose one with greatest decreasing, break tie: (subscore) sub-structure in graph • Else choose one in a random unsat clause, break tie: score>subscore>time_stamp Refinement: 1. score: clause weight; 2. subscore: different sub-structure; 3. noise strategy

3/3/2021 17 of 45 Constructive Lower Bound SAT Solving Some Results Previous Best Our Result Genetic algorithm: 3 days 25 mins Simulated Annealing: hours 20 secs Simulated Annealing: hours 15 secs Unproved 138 hrs

3/3/2021 18 of 45 Constructive Lower Bound SAT Solving

3/3/2021 19 of 45 Lower Bound Recurrence General

3/3/2021 20 of 45 Lower Bound Recurrence relation • Needs base values • Far from tight? • No explicit graph? Comparison SAT solving • Doesn’t need • Close to upper bound? • Constructive graph built

3/3/2021 21 of 45 Lower Bound Recurrence Comparison Recurrence relation • Need base value • Might far from tight • No explicit graph SAT solving • Don’t need • Close to upper bound • Constructive graph built

3/3/2021 22 of 45 Lower Bound Recurrence Relation SAT solving can provide better base values for recurrence!

3/3/2021 23 of 45 Lower Bound Recurrence Form

3/3/2021 24 of 45 Lower Bound Recurrence Proof procedure: Pasting Step 1: Copy

3/3/2021 25 of 45 Lower Bound Recurrence Proof Example: Pasting Step 1: Copy

3/3/2021 26 of 45 Lower Bound Recurrence Proof Example: Pasting Step 1: Copy

3/3/2021 27 of 45 Lower Bound Recurrence Proof Example: Pasting Step 1: Copy

3/3/2021 28 of 45 Lower Bound Recurrence Proof Example: Pasting Step 1: Copy

3/3/2021 29 of 45 Lower Bound Recurrence Proof Example: Pasting Step 1: Copy Step 2: Paste

3/3/2021 30 of 45 Lower Bound Recurrence Proof Example: Pasting Step 1: Copy Step 2: Paste Claim: • No red clique of size 6 • No blue clique of size 5